I have the active rear differential. It was a $500 option as I recall - quite a bargain (the "heavy duty package.") I can't control it in any way - it is fully automatic and integrated into the terrain response system.
The full size rear tire could hang in some off road situations, although it is tucked up pretty good into the under carriage. A bigger problem would probably be accessing it in order to change out a flat - there are a lot of off road situations where getting under the truck could be difficult. Also, the spare tire well is a very tight fit, and a larger tire (if you went aftermarket) would either have to be somewhat deflated if it would fit at all. There are aftermarket swing away tire mounts for the rear deck lid though, if this concerns you.
As an upside, the swing away tire mount helps dress up the rather bland rear of the truck (the only real styling shortfall on the vehicle, in my opinion.) You can also then get one of those cool rear tire covers with a nice rhino or lion on it. Always liked that look.