I agree that a month is a short period of time to do all that you plan. We're a big country. If you don't have a time limit, might be good to slow down a bit.
We rented a small class C RV for a week from Cruise America 7-8 years ago. Expensive but got the job done.
We met a couple on the east coast once who were campground owners from England. They had spent 8 weeks crossing the country in a Class C they purchased in California and were shipping it back to England. Buying a good used RV, then selling it before you leave, might be another option.
As for rules and expectations here, I would find a good, comprehensive book to use as a resource, and add to it/get answers to questions as you get farther into this.