Can you create your own form with 2CheckOut or do you have to link to an external page. I would like to create my own form that allows 2CheckOut to process the payment in the background. Is this possible?

You link to an external 2checkout page.
They collect the customer's credit card information and address on their page.

Nop your can't do "payment in the background" because client must enter their details on 2co payment page. you can make a forum that ask client details then forward to 2checkout then you can get passback.

Also read this https://sellers.2checkout.com/language_guidelines.html :)