At most Caribbean Islands anchoring near a town is quite feasible... and usually in reasonable (12-25 feet) depths. It is, of course, cheaper but care should be taken not to damage coral, find a sandy patch for the hook.
A few Islands don't have terrific harbours, either due to depth or lack of protection from swells, and anchoring can be quite uncomfortable. In many such cases mooring buoys are available for a nominal fee.
Many of the hotels/resorts have dinghy docks, or the town itself has a central public dock of some kind. A lot of people lock their dinghies and motors while ashore, others hire locals to "keep and eye" on their gear. In many Islands there is a well organized system of Boat Boys, who will assist you in finding a buoy, or a good anchoring spot, and for a smallish fee this generates a lot of goodwill and keeps the local economy going. Some are a bit pushy, esp those who try to sell trinkets and services you may not want. But try not to offend them.
We did not sleep on deck, mostly due to the inevitable overnight downpours (only a few minutes, but usually a couple/night and hard)
A dodger would not be required, and we've sailed hundreds of miles without even a Bimini, but after a recent trip to Mexico WITH a bimini, I'd agree that one is indispensable. Without one you're really at risk of serious sunburn, and the first job after dropping the hook is to rig a cockpit shade of some sort. I'm not a sunscreen person, but down there it's not an option.
Bugs are rare because the wind rarely dies down, but occasionally it does and some areas can get bad.
Cuba would be safe for me, but I'm Canadian....
Haven't sailed one of those charter cats, but have observed that they are not a whole lot faster than a well sailed monohull in those conditions.
One more thing: my comments are based primarily on the Caribbean chain east and south of Antigua/St Maarten down to Grenada, not the BVI, or Bahamas.. we've not been there yet.
1984 Fast/Nicholson 345 "FastForward"
".. there is much you could do at sea with common sense.. and very little you could do without it.."
Capt G E Ericson (from "The Cruel Sea" by Nicholas Monsarrat)