Should I create a new Form or use form alter?
The distinction is simple:


If you need to add some fields to an existing form, and keep all of it's original functionality (or at least most of it), use hook_form_alter.
If you need to create a totally new experience, put it under different path and create form from scratch, like shown in Examples for developers
If you feel you need totally new form under already occupied path, think again as it might cripple existing functionality.Feel free to ask more specific question then.


Then, if you want a form at 2 pages, you have 2 choices:


If form will be the only significant element on these pages, make it a page (or pages) with hook_menu 
If there are supposed to be other forms, submitted independently, wrap it into a block.