It can have it's own members area look if you setup a membership for it. It will have it's own look, however, the members area would share the domain from the primary site that the piggyback is attached to.
If an existing Member purchased into a separate membership if they use the same email, then they would cancel their old subscription and would belong to the new membership. If they use a new email and have 2 separate accounts, then they would have 2 separate logins and see separate things in each members area.
You can create optin forms using HTML form creator.
You can add custom fields and copy/paste the code to your 3rd party website.