I haven't been on Disney, but seeing as how it is a Disney cruise, I would think there would be plenty for your 7 year old to do without getting bored.
As is the case with BINGO on Royal Caribbean, if a kid wins at BINGO, then an adult must claim the prize, as bingo is still legally a form of gambling.
As I said, I haven't been on Disney, and don't have kids, so I probably can't be much help. I did however, go to Nassau a few years ago. If you like sightseeing, get a cab and go over to the Atlantis Resort. I think it's about a $4.00 cab fare one way if memory serves me correctly. Maybe someone that has been there recently can give an update on the fare.
Not knowing how Disney operates, I can't give you an estimate on how much to pay for things onboard. I know Royal Caribbean uses a cashless system, but don't know if Disney operates the same way.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)