I think the most important point in this discussion is that the individual state's laws apply as soon as you cross the border. There is no single nationwide law that applies. You have to conform to the local law in EACH and EVERY state you enter, so be sure you know the law before you cross the state border. Individual cities, tows, counties may have further restrictions. And yes, some Rv parks, whether private or public, prohibit guns. Most federal parks do, e.g. National Parks, Corp of Engineer parks, etc.
As for taking hand guns to Mexico or Canada, forget it. No dice and penalties are severe.
Generally speaking, your RV is considered a "home" for legal purposes while you are using it as a residence, in the same sense that a hotel room or an apartment is a temporary home. The status is less clear while it is in motion and is subject to interpretation the the courts, depending on circumstances and local law.
There are booklets on state laws available for those who travel with fire arms - one such can be found at http://rveducation101.com/detailpages/Firearms.htm?SiteID=1