Yes, self parking is available with your room rate and the GC, at least when I went in 2002. You park over across the street in the Paradise Pier's parking lot. I know it is pretty inconvenient when you do need your car, but the location and convenience of the hotel to both Disney parks cannot be beat! I noticed that the more expensive hotels have their parking lots very far away.
I am not sure about the laundry because we never had to use it, but I am pretty sure that they would have one?? If the GF does, then I'll bet the GC would. Just my 2 cents though. I LOVE that hotel most of all, it is my absolute favorite of all the Disney hotels I have stayed at. Have a great time!!!
I never use parking, so I can't speak to that. I haven't used the laundry, either--we save up old ratty socks and underwear for vacation, then throw them away as we go!!!!!!
However, you will have an absolute blast. I love the GCH.