I have not purchased commerce yet but I'm wondering if it's possible to, after purchase of a member group subscription I could send them to a custom page? (This page would then sync that group with my game server)

Not exactly, however you could create a hook on the onPaid() method of the Nexus product extension to send a request to your game server when a product is purchased. This would probably be a smoother workflow and experience for the user than being redirected to a third party page anyways.