I was on the Majesty for my first cruise back in 2007. I think that's a good choice for your first cruise. A short cruise, and a relatively small ship compared to the others in RCI's fleet.
Yes, there is lots to do during the times you are at sea. At least one pool will be open 24 hours. I don't believe the water is heated, but I could be wrong. It may feel good, since it may be hot and humid when you're on your cruise. The bars and lounges also sell soda, so as Dave mentioned, your daughter won't have a problem getting into the lounges; she just can't buy alcohol.
Also, regarding Bingo nights, your daughter cannot claim a prize, as Bingo is still considered a form of gambling, and you must be 21. Your daughter can play, but I believe if she wins, then you must claim her prize.
There are also shows each night in the auditorium / theater either before or after dinner, depending on what seating you have for dinner (main seating, second seating, or My Time Dining, or whatever RCI calls it). You do not have to attend these shows, just something to keep in mind.
Enjoy! Don't say you weren't warned if you come back from your cruise, and are hooked!
2007 - Majesty of the Seas
2009 - Enchantment of the Seas
2010 - Freedom of the Seas