Single Sign On between Wordpress and a Custom Website and how to do it?

Can we have a custom site (developed in php), have a Single Sign On with a Wordpress Membership site hosted as a subdomain on this custom site? If so, how?

What are the best ways to add a membership area to Wordpress? Best addons? We need a membership area to show content, videos, info products etc after they make a purchase.

What payment gateway is recommended for the above?