custom post type template changes between preview and publish

Question

I have set up a custom post titled project where I have also set a theme file page-project.php. When I create a post under that custom post type, and I preview it, it shows on the using the page-project.php template, however once I publish it and view the page, it uses the index.php template. Can someone tell me why this is happening?

This is my custom post type from functions.php

FYI: just found this out, if I turn permalinks to default settings, it uses the correct template as well.

add_action('init', 'project_register'); 
function project_register() {
    $labels = array(
        'name' => _x('My Projects', 'post type general name'),
        'singular_name' => _x('Project', 'post type singular name'),
        'add_new' => _x('Add New', 'project item'),
        'add_new_item' => __('Add New Project Item'),
        'edit_item' => __('Edit Project Item'),
        'new_item' => __('New Project Item'),
        'view_item' => __('View Project Item'),
        'search_items' => __('Search Project'),
        'not_found' =>  __('Nothing found'),
        'not_found_in_trash' => __('Nothing found in Trash'),
        'parent_item_colon' => ''
    );
    $args = array(
        'labels' => $labels,
        'public' => true,
        'publicly_queryable' => true,
        'show_ui' => true,
        'query_var' => true,
        'menu_icon' => get_stylesheet_directory_uri() . '/article16.png',
        'rewrite' => true,
        'capability_type' => 'page',
        'hierarchical' => false,
        'menu_position' => null,
        'supports' => array('title','editor','thumbnail','author')
      );

    $labels = array(
    'name' => _x( 'Project Types', 'taxonomy general name' ),
    'singular_name' => _x( 'Project Type', 'taxonomy singular name' ),
    'search_items' =>  __( 'Search Project Types' ),
    'all_items' => __( 'All Project Types' ),
    'parent_item' => __( 'Parent Project Type' ),
    'parent_item_colon' => __( 'Parent Project Type:' ),
    'edit_item' => __( 'Edit Project Type' ), 
    'update_item' => __( 'Update Project Type' ),
    'add_new_item' => __( 'Add New Project Type' ),
    'new_item_name' => __( 'New Project Type Name' ),
    'menu_name' => __( 'Project Type' ),
  ); 
    register_taxonomy('project_type',array('project'), array('hierarchical' => true,
    'labels' => $labels,
    'show_ui' => true,
    'query_var' => true,
    'rewrite' => array( 'slug' => 'type' ),));
    register_post_type( 'project' , $args );
}

Answer 1

Custom Posts don't use custom static page template files. Custom Posts have their own template files for custom post archive index pages and single custom post pages.

For a single custom post page, you need to create a single-{post-type}.php, which for you would be single-project.php.

Similarly, for a custom post archive index page, you need to create a archive-{post-type}.php, which for you would be archive-project.php.