How can I have multiple versions of the same form, in different pages? There is a default Drupal 7 form called "user_profile_form". I would like to show certain elements of this form on one page of my website, and then a different set of elements from this same form on another page.

Is there a way to either:

a) have multiple versions of the same form (keeping the same ID) or
b) clone the form and then modify it (there will be two different form IDs)

Can you point me in the right direction?
You can use hook_menu() with drupal_get_form() to use any form on any path (see Using a node editing form on a page that not a node edit page). 

You can then use hook_form_alter(), and get that to work differently depending on the path. 

You can use $_GET['q'] for the full path or arg() to get parts of the path. and use if() logic to determine how to alter the form.