So you are thinking to just do an easy drive? No stopping or touristy things? Besides being a lot of driving, you will be seeing things all over the place that you may want to stop and check out. You may want to think about including extra days for sight seeing.
Having that many pets... Tell campgrounds you have 3 dogs, and the cats will be staying in the camper (I am assuming the cats are staying inside). I don't think that will be an issue - I think most camp grounds are worried about dogs running around, not pets that stay inside. Also! Even campgrounds that say "no pets" may let you overnight on a cross country trip! So make sure you call and ask. You might be "stuck" in an overflow parking area, but you'll have access to showers and laundry machines.
Good luck, see you on the road!
