I just got back from a week on a live-aboard in Turks & Caicos and I can tell you that the diving is very very nice. We went to the 3 places that you mentioned. I just posted some pictures to whet your appetite.
Venture Explorers Live-Aboard in Turks & Caicos
Regarding whether 3 days is enough to learn to dive. The answer is no. It is enough to start your education, but you will continue to be learning for several years after as well and booking dives for the next 4 days is a perfect way to do that. Diving in such a nice place with so few dives is a double edged sword. You will be utterly amazed at everything you see but you will not have the experience yet to feel completely comfortable.
On the live-aboard last week, there was a young woman who had only 4 dives before and doubled her dives in the first day!
She enjoyed it immensely but made some decisions that I would not have made (like going to her Nitrox max depth, just to go deep).
I would stick with one operator and stay with the DM at all times. If you are going with a buddy, make sure you practice good buddy skills.
Don't push yourself and take some time during the dives to practice your skills. Go slow.. resist the urge to zoom around. You will see more if you stay still or go very very slow and wait for things to come to you.
BTW, you will probably see sharks on some of your dives in T&C, so be prepared. They are fun to be around.