(Edit: going for the "Student" badge, and wondering if anyone out there could up vote this question?)

// 1.)  Works as expected if user has every custom capability
add_action('init', function() {
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => TRUE,    // Take note of this
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
}

// 2.)  Same as #1 with the exception that access to 'post-new.php' when "Add New" button is clicked is prohibited
add_action('init', function() {
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => 'my-menu-item',  // Take note of this
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
}
add_action('admin_menu', function() {
    if(current_user_can('special-subscriber')
    ) {
        add_menu_page(
            'CPT in title bar',
            'CPT in menu',
            'edit_subs',
            'my-menu-item',
            ''
        );
    }
}

// 3.)  Breaks the same as #2
add_action('init', function() {
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => FALSE,   // Take note of this
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
}
add_action('admin_menu', function() {
    global $submenu;
    if(current_user_can('special-subscriber')
    ) {
        add_menu_page(
            'CPT in title bar',
            'CPT in menu',
            'edit_subs',
            'my-menu-item'
        );
        add_submenu_page(
            'my-menu-item',
            get_post_type_object('sub-cpt')->label,
            get_post_type_object('sub-cpt')->label,
            'edit_subs',
            'my-menu-item-sub'
        );
        // Change link
        $url = 'edit.php?post_type=sub-cpt';
        $submenu['my-menu-item'][1][2] = admin_url($url);
         unset($submenu['my-menu-item'][0]);
     }
});

// 1.)  Works as expected if user has every custom capability
add_action('init', function() {
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => TRUE,    // Take note of this
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
}

// 2.)  Same as #1 with the exception that access to 'post-new.php' when "Add New" button is clicked is prohibited
add_action('init', function() {
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => 'my-menu-item',  // Take note of this
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
}
add_action('admin_menu', function() {
    add_menu_page(
        'CPT in title bar',
        'CPT in menu',
        'edit_subs',
        'my-menu-item',
        ''
    );
}

// 3.)  Breaks the same as #2
add_action('init', function() {
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => FALSE,   // Take note of this
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
}
add_action('admin_menu', function() {
    global $submenu;
    add_menu_page(
        'CPT in title bar',
        'CPT in menu',
        'edit_subs',
        'my-menu-item'
    );
    add_submenu_page(
        'my-menu-item',
        get_post_type_object('sub-cpt')->label,
        get_post_type_object('sub-cpt')->label,
        'edit_subs',
        'my-menu-item-sub'
    );
    // Change link
    $url = 'edit.php?post_type=sub-cpt';
    $submenu['my-menu-item'][1][2] = admin_url($url);   // Set URL to view CPT
    unset($submenu['my-menu-item'][0]);             // Remove WP generated menu item
});

I have two custom post types both fully accessible to Administrators. I have Subscribers, some having access to one of these CPT's which is a "child" of the first and stores it's parent's ID in its _adm_id metadata. These "special" Subscribers have access to the parent CPT admin table so they can click a link to create child CPT posts of parents with a special status. Next, the Subscriber is allowed to edit child posts (both its own and those created by others) but only if it is of a particular custom post status. Lastly, special Subscribers are not allowed to delete posts (or edit deleted posts), not even their own.

Here's what I've got (working code) ...

// Setup custom post types and statuses
add_action('init', function() {
    // Custom Post Types
    register_post_type('adm-cpt', array(
        'label'             => __('Admin Only CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => 'my-menu-item',
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'adm',
        'map_meta_cap'      => TRUE,
        'capabilities' => array(
            'create_posts'  => 'administrator', // Only admin can create, not special Subscribers
        ),
    ));
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => 'my-menu-item',
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
    // Custom Post Statuses
    foreach(array(
        'adm-childable'     => __('Can Create Children'),
        'sub-editable'      => __('Any Subscriber Can Edit'),
    ) as $slug => $label) {
        register_post_status($slug, array(
            'label'         => _x($label, 'post'),
            'label_count'   => _n_noop($label .' <span class="count">(%s)</span>', $label .' <span class="count">(%s)</span>' ),
            'public'        => TRUE,
        ));
    }
});

// Setup parent page in admin menu
add_action('admin_menu', function() {
    // Add menu item
    if(current_user_can('administrator')
    || current_user_can('special-subscriber')
    ) {
        // Admin menu header
        add_menu_page(
            NULL,
            'CPTs',
            'exist',
            'my-menu-item',
            ''
        );
    }
});

// Set up role
add_action('wp_roles_init', function($wp_roles){
    // Prepare
    $role = 'special-subscriber';
    $caps = array(
        'delete_subs'           => FALSE,   // No trashing ...
        'delete_others_subs'    => FALSE,
        'delete_published_subs' => FALSE,
        'delete_private_subs'   => FALSE,
        'edit_published_subs'   => FALSE,   // And no editing published/private posts ...
        'edit_private_subs'     => FALSE,
        'edit_adms'             => TRUE,    // Allow viewing of adm-cpt table
        'edit_posts'            => TRUE,    // WARNING:  Here's the permission that is causing the problems!
    );
    $name = __('"Special" Subscriber');
    // Update role in database, if needed
    if($wp_roles->get_role($role) === NULL
    || $wp_roles->get_role($role)->capabilities != $caps
    || $wp_roles->roles[$role]['name'] !== $name
    ) {
        $wp_roles->remove_role($role);
        $wp_roles->add_role($role, $name, $caps);
    }
});

// Dynamicly set capabilities
add_action('user_has_cap', function($allcaps, $caps, $args, $user) {
    foreach($caps as $cap) {
        $perm = substr($cap, 0, strrpos($cap, '_'));
        $type = substr($cap, strlen($perm)+1);
        if(in_array($type, array('adm', 'adms')) && in_array('administrator', $user->roles)
        || in_array($type, array('sub', 'subs')) && !empty(array_intersect(array('administrator', 'special-subscriber'), $user->roles))
        ) {
            // Check Subscriber if post is editable
            if(in_array($cap, array('edit_subs', 'edit_others_subs'))
            && in_array('special-subscriber', $user->roles)
            && !in_array('administrator', $user->roles)
            && !empty($args[2])
            &&  (   !in_array(get_post_status($args[2]), array('sub-editable'))
                        && !in_array($_REQUEST['original_post_status'], array('sub-editable', 'auto-draft'))    // Creating
                ||  get_post_status(get_post_meta($args[2], '_adm_id', TRUE)) === 'trash'
                )
            ) {
                $allcaps[$cap] = FALSE;
            }
            // Add the cap
            if(!isset($allcaps[$cap])
            ) {
                $allcaps[$cap] = TRUE;  // All the _adm and _sub capabilities are made available.
            }
        }
    }
    return $allcaps;
}, 10, 4);

// Add stuff to force proper navigation
add_action('post_row_actions', function($actions, $post) {
    // Add link to adm-cpt table entries to create child
    if(get_post_type($post) === 'adm-cpt'
    && get_post_status($post) === 'adm-childable'
    && current_user_can('edit_subs')
    ) {
        $lbl = __('New '). get_post_type_object('sub-cpt')->labels->name;
        $actions['adm-cpt-create-sub-cpt'] = sprintf(
            '<a href="%s" aria-label="%s">%s</a>',
            admin_url('post-new.php?post_type=sub-cpt&adm_id='. $post->ID),
            esc_attr('“'. $lbl .'”'),
            $lbl
        );
    }
    // Return
    return $actions;
}, 10, 2);

// Modify publish metabox
add_action('post_submitbox_misc_actions', function($post) {
    $arr = array();
    switch(get_post_type($post)) {
        case 'adm-cpt':
            $arr = array('adm-childable');
            break;
        case 'sub-cpt':
            $arr = array('sub-editable');
            break;
        default:
            return;
    }
    // Check that parent exists -- Should be in an init hook, but it's prettier here.
    if($_REQUEST['post_type'] === 'sub-cpt'
    &&  (empty($_REQUEST['adm_id']) || get_post_type($_REQUEST['adm_id']) !== 'adm-cpt')
    &&  (empty($post->_adm_id) || get_post_type($post->_adm_id) !== 'adm-cpt')
    ){
        ?><script>window.document.location.replace("<?= admin_url('edit.php?post_type=adm-cpt') ?>")</script><?php
        return;
    }
    // Add custom post statuses
    ?><input type='hidden' name='adm_id' value='<?= $_REQUEST['adm_id'] ?>'><?php
    if(count($arr)) {
        ?><script>
        <?php foreach($arr as $k) { $obj = get_post_status_object($k); ?>
            jQuery("select#post_status").append("<option value=\"<?= $k ?>\"><?= $obj->label ?></option>");
            <?php if(get_post_status($post) == $k) { ?>
                jQuery("#post-status-display").text("<?= $obj->label ?>");
                jQuery("select#post_status").val("<?= $k ?>");
            <?php } ?>
        <?php } ?>
        </script><?php
    }
    // Display parent -- Informational
    if(!empty($_REQUEST['adm_id'])
    || !empty($post->_adm_id)
    ) {
        $parent_id = $post->_adm_id;
        if(!$parent_id) $parent_id = $_REQUEST['adm_id'];
        ?><div class="misc-pub-section misc-pub-adm-cpt">Parent:  <span id="post-status-display"><?= get_the_title($parent_id) ?></span></div><?php
    }
});

// Save parent ID
add_action('save_post_sub-cpt', function($post_id, $post, $update) {
    // Ensure we continue only id a new child is created
    if(defined('DOING_AUTOSAVE') && DOING_AUTOSAVE
    || get_post_type($post_id) !== 'sub-cpt'
    || empty($_REQUEST['adm_id'])
    || get_post_type($_REQUEST['adm_id']) !== 'adm-cpt'
    ) return;
    // Set parent ID
    update_post_meta($post_id, '_adm_id', $_REQUEST['adm_id']);
}, 10, 3);

// Navigation when changed to uneditable
add_action('load-post.php', function(){
    if(!empty($_REQUEST['post'])
    && get_post_type($_REQUEST['post']) === 'sub-cpt'
    && !current_user_can('edit_subs', $_REQUEST['post'])
    ) {
        delete_post_meta($_REQUEST['post'], '_edit_lock');
        wp_redirect('edit.php?post_type=sub-cpt');
        die();
    }
});

This issue here is that the special Subscriber is able to edit regular Posts and Comments. I understand this comes from the edit_posts capability, and that capability allows for the editing/creation of all post types. However, removing it prevents special Subscribers from being able to create sub-cpt posts, and granting edit_subs does not solve the problem. Neither does setting the capabilities->create_post=special-subscriber when registering the child CPT. I've been able to limit the ability of Subscribers from being able to create adm-cpt posts by defining the capabilities parameter when registering the post type. But I don't want special Subscribers to be able to edit/create any other posts other than those of the sub-cpt type, and I can't seem to figure out how.

I've found a Q&A related to the subject, but this doesn't seem to work. The CPT's are mapped to custom capabilities, they exist, and the user_has_cap filter dynamically grants each of these capabilities. I've even tried expressly defining them in the special-subscriber role definition. Anyways, I'm sure the change is simple--what is it?

(If you're interested, I have another capability problem. When a special Subscriber sets the child CPT post_status to publish, the post is locked and they are forwarded to edit.php but I want the post to unlock and for the viewer to be forwarded to edit.php?post_type=sub-cpt just like is done in the load-post.php hook of my code, and I can't seem to figure out how.)

I have two custom post types both fully accessible to Administrators. I have Subscribers, some having access to one of these CPT's which is a "child" of the first and stores it's parent's ID in its _adm_id metadata. These "special" Subscribers have access to the parent CPT admin table so they can click a link to create child CPT posts of parents with a special status. Next, the Subscriber is allowed to edit child posts (both its own and those created by others) but only if it is of a particular custom post status. Lastly, special Subscribers are not allowed to delete posts (or edit deleted posts), not even their own.

// Setup custom post types and statuses
add_action('init', function() {
    // Custom Post Types
    register_post_type('adm-cpt', array(
        'label'             => __('Admin Only CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => 'my-menu-item',
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'adm',
        'map_meta_cap'      => TRUE,
        'capabilities' => array(
            'create_posts'  => 'administrator', // Only admin can create, not special Subscribers
        ),
    ));
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => 'my-menu-item',
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
    // Custom Post Statuses
    foreach(array(
        'adm-childable'     => __('Can Create Children'),
        'sub-editable'      => __('Any Subscriber Can Edit'),
    ) as $slug => $label) {
        register_post_status($slug, array(
            'label'         => _x($label, 'post'),
            'label_count'   => _n_noop($label .' <span class="count">(%s)</span>', $label .' <span class="count">(%s)</span>' ),
            'public'        => TRUE,
        ));
    }
});

// Setup parent page in admin menu
add_action('admin_menu', function() {
    // Add menu item
    if(current_user_can('administrator')
    || current_user_can('special-subscriber')
    ) {
        // Admin menu header
        add_menu_page(
            NULL,
            'CPTs',
            'exist',
            'my-menu-item',
            ''
        );
    }
});

// Set up role
add_action('wp_roles_init', function($wp_roles){
    // Prepare
    $role = 'special-subscriber';
    $caps = array(
        'delete_subs'           => FALSE,   // No trashing ...
        'delete_others_subs'    => FALSE,
        'delete_published_subs' => FALSE,
        'delete_private_subs'   => FALSE,
        'edit_published_subs'   => FALSE,   // And no editing published/private posts ...
        'edit_private_subs'     => FALSE,
        'edit_adms'             => TRUE,    // Allow viewing of adm-cpt table
        'edit_posts'            => TRUE,    // WARNING:  Here's the permission that is causing the problems!
    );
    $name = __('"Special" Subscriber');
    // Update role in database, if needed
    if($wp_roles->get_role($role) === NULL
    || $wp_roles->get_role($role)->capabilities != $caps
    || $wp_roles->roles[$role]['name'] !== $name
    ) {
        $wp_roles->remove_role($role);
        $wp_roles->add_role($role, $name, $caps);
    }
});

// Dynamicly set capabilities
add_action('user_has_cap', function($allcaps, $caps, $args, $user) {
    foreach($caps as $cap) {
        $perm = substr($cap, 0, strrpos($cap, '_'));
        $type = substr($cap, strlen($perm)+1);
        if(in_array($type, array('adm', 'adms')) && in_array('administrator', $user->roles)
        || in_array($type, array('sub', 'subs')) && !empty(array_intersect(array('administrator', 'special-subscriber'), $user->roles))
        ) {
            // Check Subscriber if post is editable
            if(in_array($cap, array('edit_subs', 'edit_others_subs'))
            && in_array('special-subscriber', $user->roles)
            && !in_array('administrator', $user->roles)
            && !empty($args[2])
            &&  (   !in_array(get_post_status($args[2]), array('sub-editable'))
                        && !in_array($_REQUEST['original_post_status'], array('sub-editable', 'auto-draft'))    // Creating
                ||  get_post_status(get_post_meta($args[2], '_adm_id', TRUE)) === 'trash'
                )
            ) {
                $allcaps[$cap] = FALSE;
            }
            // Add the cap
            if(!isset($allcaps[$cap])
            ) {
                $allcaps[$cap] = TRUE;  // All the _adm and _sub capabilities are made available.
            }
        }
    }
    return $allcaps;
}, 10, 4);

// Add stuff to force proper navigation
add_action('post_row_actions', function($actions, $post) {
    // Add link to adm-cpt table entries to create child
    if(get_post_type($post) === 'adm-cpt'
    && get_post_status($post) === 'adm-childable'
    && current_user_can('edit_subs')
    ) {
        $lbl = __('New '). get_post_type_object('sub-cpt')->labels->name;
        $actions['adm-cpt-create-sub-cpt'] = sprintf(
            '<a href="%s" aria-label="%s">%s</a>',
            admin_url('post-new.php?post_type=sub-cpt&adm_id='. $post->ID),
            esc_attr('“'. $lbl .'”'),
            $lbl
        );
    }
    // Return
    return $actions;
}, 10, 2);

// Modify publish metabox
add_action('post_submitbox_misc_actions', function($post) {
    $arr = array();
    switch(get_post_type($post)) {
        case 'adm-cpt':
            $arr = array('adm-childable');
            break;
        case 'sub-cpt':
            $arr = array('sub-editable');
            break;
        default:
            return;
    }
    // Check that parent exists -- Should be in an init hook, but it's prettier here.
    if($_REQUEST['post_type'] === 'sub-cpt'
    &&  (empty($_REQUEST['adm_id']) || get_post_type($_REQUEST['adm_id']) !== 'adm-cpt')
    &&  (empty($post->_adm_id) || get_post_type($post->_adm_id) !== 'adm-cpt')
    ){
        ?><script>window.document.location.replace("<?= admin_url('edit.php?post_type=adm-cpt') ?>")</script><?php
        return;
    }
    // Add custom post statuses
    ?><input type='hidden' name='adm_id' value='<?= $_REQUEST['adm_id'] ?>'><?php
    if(count($arr)) {
        ?><script>
        <?php foreach($arr as $k) { $obj = get_post_status_object($k); ?>
            jQuery("select#post_status").append("<option value=\"<?= $k ?>\"><?= $obj->label ?></option>");
            <?php if(get_post_status($post) == $k) { ?>
                jQuery("#post-status-display").text("<?= $obj->label ?>");
                jQuery("select#post_status").val("<?= $k ?>");
            <?php } ?>
        <?php } ?>
        </script><?php
    }
    // Display parent -- Informational
    if(!empty($_REQUEST['adm_id'])
    || !empty($post->_adm_id)
    ) {
        $parent_id = $post->_adm_id;
        if(!$parent_id) $parent_id = $_REQUEST['adm_id'];
        ?><div class="misc-pub-section misc-pub-adm-cpt">Parent:  <span id="post-status-display"><?= get_the_title($parent_id) ?></span></div><?php
    }
});

// Save parent ID
add_action('save_post_sub-cpt', function($post_id, $post, $update) {
    // Ensure we continue only id a new child is created
    if(defined('DOING_AUTOSAVE') && DOING_AUTOSAVE
    || get_post_type($post_id) !== 'sub-cpt'
    || empty($_REQUEST['adm_id'])
    || get_post_type($_REQUEST['adm_id']) !== 'adm-cpt'
    ) return;
    // Set parent ID
    update_post_meta($post_id, '_adm_id', $_REQUEST['adm_id']);
}, 10, 3);

// Navigation when changed to uneditable
add_action('load-post.php', function(){
    if(!empty($_REQUEST['post'])
    && get_post_type($_REQUEST['post']) === 'sub-cpt'
    && !current_user_can('edit_subs', $_REQUEST['post'])
    ) {
        delete_post_meta($_REQUEST['post'], '_edit_lock');
        wp_redirect('edit.php?post_type=sub-cpt');
        die();
    }
});

(If you're interested, I have another capability problem. When a special Subscriber sets the child CPT post_status to publish, the post is locked and they are forwarded to edit.php but I want the post to unlock and for the viewer to be forwarded to edit.php?post_type=sub-cpt just like is done in the load-post.php hook of my code, and I can't seem to figure out how.)

UPDATE: I've isolated it down to the placement of the CPT in the menu. When the CPT is registered as showing the UI using the register_post_type option of show_in_menu=TRUE, everything works as expected. But, when the CPT is added as a submenu of an old-fashioned admin menu item, things break. Adding the UI and hiding it results in the same problems, along with adding a subpage and redirecting it to the UI of the CPT. Examples:

// 1.)  Works as expected if user has every custom capability
add_action('init', function() {
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => TRUE,    // Take note of this
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
}

// 2.)  Same as #1 with the exception that access to 'post-new.php' when "Add New" button is clicked is prohibited
add_action('init', function() {
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => 'my-menu-item',  // Take note of this
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
}
add_action('admin_menu', function() {
    if(current_user_can('special-subscriber')
    ) {
        add_menu_page(
            'CPT in title bar',
            'CPT in menu',
            'edit_subs',
            'my-menu-item',
            ''
        );
    }
}

// 3.)  Breaks the same as #2
add_action('init', function() {
    register_post_type('sub-cpt', array(
        'label'             => __('Subscriber/Admin CPT'),
        'show_ui'           => TRUE,
        'show_in_menu'      => FALSE,   // Take note of this
        'show_in_admin_bar' => FALSE,
        'capability_type'   => 'sub',
        'map_meta_cap'      => TRUE,
    ));
}
add_action('admin_menu', function() {
    global $submenu;
    if(current_user_can('special-subscriber')
    ) {
        add_menu_page(
            'CPT in title bar',
            'CPT in menu',
            'edit_subs',
            'my-menu-item'
        );
        add_submenu_page(
            'my-menu-item',
            get_post_type_object('sub-cpt')->label,
            get_post_type_object('sub-cpt')->label,
            'edit_subs',
            'my-menu-item-sub'
        );
        // Change link
        $url = 'edit.php?post_type=sub-cpt';
        $submenu['my-menu-item'][1][2] = admin_url($url);
        unset($submenu['my-menu-item'][0]);
    }
});

If, I can get the "Add New" functionality to work with the CPT as a subpage, I think my problem will be solved because the edit_posts capability giving me trouble can be specifically mapped to edit_subs. Anyone know how to do this?