I tried using those suction cups to take out my contacts, but it got stuck and I had to peel it off. Did this damage my eye membrane?

Probably not... It is unlikely that there was permanent damage. If there was significant pain afterwards for at least a few hours, you may have scratched the cornea or caused some of the skin to come off. The cornea has the fastest regenerating skin in the body, and any scratches heal quickly. See an eye doctor if you either have pain or decreased vision (as well as for reassurance!).
Depends. It depends on what you mean by "eye membrane". The contact lens sits on your cornea and yes, undue force may cause a corneal abrasion. This is painful. If you mean the lens or retina, you are unlikely to have damaged these structures as they are internal to the eye. If you have pain, vision loss or any concerns, see your ophthalmologist.