Caye Caulker is a really relaxed place and right now most of the hotels have a lot of openings so you'll have no problem getting a room... We stayed at Tropical Paradise and Beyond the Sea... Both were very cheap (TP was $45 belize a night, BTS was $25 belize a night).. BTS was much closer to Frenchies, so we stayed with them for the shorter walk. TP is right next to Big Fish, so if you want to use them its more convenient to stay over there.
Frenchies was great, their staff is really laid back and fun to hang out with, the boat rides in between dives was always a lot of fun, and once we were under water they always found fun stuff to show us. I definitely recommend them.
When you land in the airport or from the water taxi there will be a lot of Taxi's offering you a ride, the Island is extremely small and any where you decide to go is walkable, so there is no need for a taxi.