I've never been at Thanksgiving, but there are quite a few people on this board that go annually over Thanksgiving. I'm sure they'll be able to answer all your questions. I can tell you that the pools are heated. Also, if the weather is nice, there's plenty to do at HH. Walk the beach, golf, rent bikes to ride around the island or join in some of the resort activities. There are usually activities for all ages provided by the recreation staff at the resort. My kids love a trip to HH as much as one to WDW (maybe even more). It is also just a great place to relax and unwind.
As far as room availability, I wouldn't know for that time of year. I do know that there are many more 2BR units than 1BR and often people can snag a 2BR when the studios & 1BRs are booked.
HTH