Welcome to the RTA Forum!

Its kind of hard to answer your questions right now, because you just don't seem to have enough nailed down. For example, your budget estimate of $10000 is potentially a little low, but much of that is going to depend upon your transportation. Your transportation will also determine where you can stay (camping isn't really an option if you are taking a bus or train)

All of the options you've laid out are within the range of possibilities, but they also cover a massive spread of the kind of trip you could have. 2 months in an RV slowly working your way through the country, would be a whole lot different than taking busses and trains from city to city and staying in hotels an hostels.

With 2 months, you've got plenty of time. Your budget will at least be workable, although you may have to make choices to make the numbers work, but I think your first step needs to be to figure out what kind of trip you want this to be.