7 weeks sounds like a great timeframe for such a trip. Yes, it's possible. Haven't yet camped in California, so I can't offer any help with CGs there, but friends who have driven it have gotten from s.e. Idaho to the Sacramento area in 10-12 hours driving. (This is assuming you still plan to visit Yellowstone & the Tetons - which I'd advise doing.)
Whitewater rafting could be done outside Jackson, Wyoming, or you might consider going home through Colorado - some dandy whitewater there, near the s.w. corner and in other locations (do some research on it - my knowledge of it is very, very general), as well as spectacular scenery.
Remember that the southwest is going to be hot. Depending on when you plan to leave, consider the average temperatures of the places you plan to see before planning your route - early summer, go south first, and finish up in the northern parks (i.e. Yellowstone, etc.), but if you're not leaving until August, go north first and finish up with the southern route.