Can’t seem to get JQuery working (wordpress plugin)

This is a common question. It plagued me when I first started because it doesn’t make any sense and it is hard to search for until you know. The problem is that WordPress bundles a version of jQuery that is set to noConflict mode. Therefore, to run scripts you must use noConflict wrappers

jQuery(document).ready(function($) {
    // Inside of this function, $() will work as an alias for jQuery()
    // and other libraries also using $ will not be accessible under this shortcut
});

Update

1. There’s no need to load a different version of jQuery. It just complicates things. Let, WordPress load it’s own version. Delete your jquery_enqueue() method.

2. Next adjust your create_admin_page() to not print the script directly. What I think is happening is that add_menu_page() runs before wp_enqueue_scripts and so jQuery isn’t available yet. Hence, error. Instead of printing the scripts, you can print them in the footer by adding a new function to the admin_print_footer_scripts hook.

So modified, your create_admin_page() would now look like this:

public function create_admin_page()
    {
        // Set class property
        $this->options = get_option( 'my_option_name' );
        add_action( 'admin_print_footer_scripts', array( $this, 'footer_scripts' ) );
        ?>
        <div class="wrap">
            <h2>Test</h2>           
            <form method="post" action="options.php">
            <?php
                // This prints out all hidden setting fields
                settings_fields( '...' );   
                do_settings_sections( '...' );
            ?>
            </form>
            <textarea></textarea>
        </div>
        <?php
    }

With your new function being:

public function footer_scripts()
    { ?>
            <script>                
                jQuery( document ).ready(function($) {
                    $( "#my_option_name" ).change(function() {
                        $.post( 
                          'classes/dummy.class.php'
                        ).success(function(x){
                            json = $.parseJSON(x);
                            $('textarea').text(json);
                        });
                    });
                });
            </script>