Custom fields added via form api not working as expected
AFAIK, hook_form_alter() does not require you to perform a drupal_render() in it. And I suspect that is why you are not getting a proper rendered for item.

Also, each custom field needs to processed (by you) on form submit.
Thus, you need to add $form['#submit'][] = 'my_submit_function_submit';, where my_submit_function_submit() handles the database inserts.