Almost there with your PHP function. No need to set the header.

function myAjaxFunc() {

    $posts = get_posts( array(
        'posts_per_page'   => -1,
        'orderby'          => 'title',
        'order'            => 'ASC',
        'post_type'        => 'my-post-type',
        'post_status'      => array( 'publish', 'draft' )
    ) );

    $list = array();
    foreach ( $posts as $post ) {
        $list[] = array(
            'id'   => $post->ID,
            'name' => $post->post_title,
            'link' => get_permalink( $post->ID ),
        );
    }
    echo json_encode( $list );
    die;
}
add_action( 'wp_ajax_nopriv_myAjaxFunc', 'myAjaxFunc' );
add_action( 'wp_ajax_myAjaxFunc', 'myAjaxFunc' );

And your Javascript:

$.ajax({
    url: "<?php bloginfo('url'); ?>/wp-admin/admin-ajax.php",
    type: "POST",
    data: "action=myAjaxFunc",
    success: function(results) {
        var posts = JSON.parse(results);
        console.log(results);
        $.each(posts, function() {
            $('#someSelect').append( $('<option></option>').text(this.name).val(this.id) );
        });
    },
    error: function() {
        console.log('Cannot retrieve data.');
    }
});

Almost there with your PHP function. No need to set the header. (Edit: Also, assuming get_posts() is actually returning results.)

function myAjaxFunc() {

    $posts = get_posts( array(
        'posts_per_page'   => -1,
        'orderby'          => 'title',
        'order'            => 'ASC',
        'post_type'        => 'my-post-type',
        'post_status'      => array( 'publish', 'draft' )
    ) );

    $list = array();
    foreach ( $posts as $post ) {
        $list[] = array(
            'id'   => $post->ID,
            'name' => $post->post_title,
            'link' => get_permalink( $post->ID ),
        );
    }
    echo json_encode( $list );
    die;
}
add_action( 'wp_ajax_nopriv_myAjaxFunc', 'myAjaxFunc' );
add_action( 'wp_ajax_myAjaxFunc', 'myAjaxFunc' );

And your Javascript:

$.ajax({
    url: "<?php bloginfo('url'); ?>/wp-admin/admin-ajax.php",
    type: "POST",
    data: "action=myAjaxFunc",
    success: function(results) {
        var posts = JSON.parse(results);
        console.log(results);
        $.each(posts, function() {
            $('#someSelect').append( $('<option></option>').text(this.name).val(this.id) );
        });
    },
    error: function() {
        console.log('Cannot retrieve data.');
    }
});

Almost there with your PHP function. No need to set the header.

function myAjaxFunc() {

    $posts = get_posts( array(
        'posts_per_page'   => -1,
        'orderby'          => 'title',
        'order'            => 'ASC',
        'post_type'        => 'my-post-type',
        'post_status'      => array( 'publish', 'draft' )
    ) );

    $list = array();
    foreach ( $posts as $post ) {
        $list[] = array(
            'id'   => $post->ID,
            'name' => $post->post_title,
            'link' => get_permalink( $post->ID ),
        );
    }
    echo json_encode( $list );
    die;
}
add_action( 'wp_ajax_nopriv_myAjaxFunc', 'myAjaxFunc' );
add_action( 'wp_ajax_myAjaxFunc', 'myAjaxFunc' );

And your Javascript:

$.ajax({
    url: "<?php bloginfo('url'); ?>/wp-admin/admin-ajax.php",
    type: "POST",
    data: "action=myAjaxFunc",
    success: function(results) {
        var posts = JSON.parse(results);
        console.log(results);
        $.each(posts, function() {
            $('#someSelect').append( $('<option></option>').text(this.name).val(this.id) );
        });
    },
    error function() {
        console.log('Cannot retrieve data.');
    }
});

Almost there with your PHP function. No need to set the header.

function myAjaxFunc() {

    $posts = get_posts( array(
        'posts_per_page'   => -1,
        'orderby'          => 'title',
        'order'            => 'ASC',
        'post_type'        => 'my-post-type',
        'post_status'      => array( 'publish', 'draft' )
    ) );

    $list = array();
    foreach ( $posts as $post ) {
        $list[] = array(
            'id'   => $post->ID,
            'name' => $post->post_title,
            'link' => get_permalink( $post->ID ),
        );
    }
    echo json_encode( $list );
    die;
}
add_action( 'wp_ajax_nopriv_myAjaxFunc', 'myAjaxFunc' );
add_action( 'wp_ajax_myAjaxFunc', 'myAjaxFunc' );

And your Javascript:

$.ajax({
    url: "<?php bloginfo('url'); ?>/wp-admin/admin-ajax.php",
    type: "POST",
    data: "action=myAjaxFunc",
    success: function(results) {
        var posts = JSON.parse(results);
        console.log(results);
        $.each(posts, function() {
            $('#someSelect').append( $('<option></option>').text(this.name).val(this.id) );
        });
    },
    error: function() {
        console.log('Cannot retrieve data.');
    }
});