jQuery Plugin to use WordPress functions in AJAX request

I’m using Chained Selects jQuery Plugin to create multiple chained selection fields. I’m using the remote version of the plugin to build the related options by means of AJAX request from a separate PHP file. So basically it works by calling the following code:

$("#field2").remoteChained("#field1", "/data/json.php");

What I would like to do

I would like to use WordPress’ functions in the PHP file (in this case: /data/json.php) to create the options dynamically. But I do not want to use mysql_query() to search through the database.

I understand that in WordPress you have to use admin-ajax.php. But I’m not sure how to integrate it with the plugin above.

Sample Case

For this example, let’s just use WordPress’ category. So the two select fields will be the category field and the sub-category field. The latter one will relate to the former one, ie. the parent category.

Note: In my actual project, I’m using PODs CMS plugin to
create a list of items from a content type (which is like custom
post type) that is related to another list of items in another
content type. These will become the options in my chained select fields.

What I have tried so far

1. The HTML:

   <?php
   wp_dropdown_categories('show_count=0&selected=-1&hierarchical=1&depth=1&hide_empty=0&exclude=1&show_option_none=Main Categories&name=field1');
   ?>
   <select name="field2" id="field2"></select>
   

2. Call the scripts (in functions.php):

   function the_scripts() {
       // The Chained Selects jQuery Plugin
       wp_enqueue_script('chained_select', get_template_directory_uri() . '/js/jquery.chained.remote.js', array('jquery'), false, true);
   
       // The script to call the action
       wp_enqueue_script('scripts', get_template_directory_uri() . '/js/scripts.js', array('jquery','chained_select'), false, true);
   
       // code to declare the URL to the file handling the AJAX request
       wp_localize_script( 'scripts', 'the_ajax_script', array( 'ajaxurl' => admin_url( 'admin-ajax.php' ) ) );
   }
   
   add_action('wp_enqueue_scripts', 'the_scripts', 100);
   

3. The code to call the action (in scripts.js):

   $(document).ready(function (){
   
       jQuery.post(
           the_ajax_script.ajaxurl,
               {
                   action : 'myajax-submit'
               }
               ,function( response ) {
                   $("#field2").remoteChained("#field1", the_ajax_script.ajaxurl);
               }
       );
   
   });
   

4. The action (in functions.php):

   function get_options() {
   
       $response[""] = "--";
   
       if ($_GET["field1"]) {
   
           $categories =  get_categories('child_of=".$_GET["field1'].'&hide_empty=0');
   
           $response[""] = "--";
           foreach ($categories as $cat) {
               $response[$cat->term_id] = $cat->cat_name;
           }
   
       };
   
       print json_encode($response);
   
       exit;
   }
   
   add_action( 'wp_ajax_myajax-submit', 'get_options' );
   add_action( 'wp_ajax_nopriv_myajax-submit', 'get_options' );
   

This whole thing doesn’t work in that the jQuery plugin isn’t doing it’s job. How can I communicate WordPress with this plugin?

Thanks in advanced.