how to put a submit button with a pre-defined submit function by calling drupal_render?
Are you returning those $render_elem from your form function, when drupal_get_form calls that form id callback? Because i see you call drupal_render_children, but drupal need to know about it in form of form api array. drupal_get_form expecting you return from your form_id function form api array of form elements, so it can do some operation like checking, if current request is POST and contains form_id of your form and some name of submit button.  

EDIT: Try add all submit buttons to your form array, but set their property #printed =&gt; true, so they won't be rendered, but drupal will be notified about those buttons and can run your submit functions.