The Catamount and Pinball take-outs are on BLM public lands. The Burns take-out is on private land, so you will need to get private landowner permission to use that site. The Dotsero take-out is CDOT and is also a public take-out.
As long as you are not a commercial or permitted group, you can camp anywhere on BLM public lands between State Bridge and Dotsero. There are developed campsites (include restrooms, picnic tables, metal fire grates) at Windy Point, Catamount, Pinball, and Lyon's Gulch.
I have not yet floated Rodeo Rapid at this level, so I can not give you any feedback about the rapid as it is right now. Hopefully, another boater can help with that.
Have fun on your trip!
BLM - Colorado River Valley Field Office