The problem is that you are not using any of AJAX helpers provided by Drupal. When you are using Form API AJAX, for example, all behaviors are re-initialised and functionalities like calendar added to new elements.
If you are not using Drupal-provided AJAX functionality, you need to manually call all behaviors with a proper context after your request is finished and DOM tree updated, probably by calling
Drupal.attachBehaviors(element)
on added DOM element.