I spend a lot of time in Iceland but unfortunately, I don't have any personal experience with camper vans as I always use 4x4 on my travels there. Nevertheless, Happy Campers is the company most people I come across use on their adventures and the one I see the most cruising around the country during colder months but off the F-roads of course.
As to the hotel near KEF airport, I normally use a couple hotels not more than 5-10 minutes away from departures/arrivals. The last one I used, last May, on a 8 room reservation, was Bed and Breakfast Keflavík Airport (http://www.booking.com/Share-8Y4RZx). Strong features are the airport distance, big and clean rooms (recently renovated), breakfast period starting at 4 a.m. and the fact that they provide shuttle transfer to the airport. Be sure to book it during your check-in.
EDIT: By the way. The camping card? I've camped in Iceland for 21 nights in the last 12 months and never used a camping card. If by any chance you need any help with camping sites, please let me know. As a landscape photographer, I know a few :D