I had no problem carrying my DSLR on any ride at WDW (though admittedly RnRC was closed last trip.) The only time I've gotten a hard time from a CM was on Tower of Terror (California Adventure version, not at MGM in Florida), when he didn't like that I had the lens cap off my PnS. I felt like I was getting scolded by a teacher in front of the class.
For Kali River Rapids, I kind of wrapped the DSLR in my bucket hat and stuck in in the covered area in the middle, and it was fine. Most camera bags seem to have zippers plus flaps & velcro, so if you close it up tight, your stuff should be well-protected even on a wet ride, assuming that you don't submerge it.
If you're going on a new ride, it's probably not a bad idea, if possible, to ride it the first time
without your camera out, just so you can concentrate 100% on the ride... I didn't do this during Expedition Everest and I still loved it, but next trip I'm going to have to try putting the camera down and just enjoy the ride.