Welcome to the RTA Forum!

As far as a timeline goes, you'll need a week just to drive fairly quickly across country. Throw in all the things you've listed that you want to do, including spending a week at Disneyworld in Florida, and I think you'd need a full month for the trip.

You can't just pull off to the side of the road to "camp" in an RV, but there are some free places you can park, like truck stops and some store parking lots - like Walmart. However, when you are using these free options, it means you won't have things like water and power, and pushing out the slide-outs of an RV is frowned upon, so you could end up being pretty cramped with all the people you are talking about.