amenity cards depend on who you are renting through. If you are renting through the resort, you pick them up on arrival. If you are renting privately or through a management company, you just have to check their policies. Note though that if you rent from anyone outside the resort in order to get amenity cards the property owner must be a member of the resort. Cost right now is $50 a week if you get them through anyone outside the resort.
supposed to be 16 to drive mopeds or golf carts, and are supposed to have a license on you when you drive.
as for day trips, kind of depends on what you are looking for. there is good golf on the island, although cheaper is to go to the course at the USMC base on Parris Island, which is about a 25-30 minute drive. One of the best military courses, and open to the public.
Hunting Island state park is the island before you get to Fripp. Downtown Beaufort is good to take a look at. You can always drive into Savannah, which is typically about a 90 minute drive. Others may have to help you on the off island stuff, as I generally prefer just to hang out on the beach.