We rented from KEF the first of this month. Don't recall if we ended up with Budget or Avis, but there are at least 5 companies right in the terminal. Booked it online ahead of time, and cars are right outside the terminal. A little hard to find our car at night (not well lit), but if you're going now, you should have much more light than we did.

Return was to a very slightly offsite depot, where they put us in a van and had us at the airport in 2-3 minutes.

The companies definitely share counter personnel, but I can't speak for if they share vehicles, but the dropoffs were all very near each other.

Get yourself a 4 wheel drive vehicle with high clearance so you can go down some of the side roads. You'll be very glad you did.