How can I have multiple versions of the same form, in different pages?
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.