Ajax with jQuery UI dialog not working

Assuming your javascript and ajax action code are otherwise correct, target admin-ajax.php with the correct path by localizing your enqueued script:

wp_localize_script(
    'your_script_handle',
    'script_data',
    array( 'admin_ajax' => admin_url( 'admin-ajax.php' ) )
);

Then in your javascript, reference that URL with:

url: script_data.admin_ajax

EDIT-

ah, missed this the first time: $wpdb->feedmanager isn’t set unless you’ve explicitly set it somewhere. $wpdb->table_name only works for native tables, as those member vars are hard-coded directly into the wpdb class. change it to a string 'wp_feedmanager'.

also note that the wp_ table prefix can be (and should be) changed via wp-config.php, use $wpdb->prefix to make your code more portable:

$table_name = $wpdb->prefix . 'feedmanager';