We were just on Carnival Valor from 6/22-6/29, and had a great time. We went zip-lining and cave tubing in Belize and it was a lot of fun. The zip-lining was the best part by far, and I highly recommend it. Cave tubing was fun; you have to walk about 40 minutes, carrying your inner tube, but it was a fairly easy walk, just a little slippery at times. The water in the cave was a little cold, but you get use to it. The caves were dark, but you have a light on your head and we did not see any snakes, bats, etc. I highly recommend Xtreme or Reggie's as they take smaller groups and you don't have to wait so long in line. The cruise excursions had bus fulls and they had to wait a long time. You should bring a bathing suit, aqua socks or water shoes, and a change of clothes. Dont' forget mosquito repellent! This was probably the highlight of all our excursions.
Yes, I agree with kwtraveler here, book through Reggie's or Xtreme, the cruise line tours are overly crowded and are not nearly as long of a float as the private guides. So you get more for less money.
As far as clastrophobic, yes, maybe a little. You will enter caves that are complete blackness except for the falsh light you have. If you are terribly clastrophobic, you might want to consider something else. But the caves do not last long.. you go in and out of them to daylight.
No "icky things" the water is basically an underground river coming out through the caves to the surface... so nothing lives in it for the part you are tubing in. The worst things are the ants on the trail going out to the starting point. But if you watch for them and don't step in them you are fine.