Hi Addio
I can't answer all your questions, but I'll take a crack at a couple of them.
Home Base - I don't know about needing a physical address for cyber school, but you don't need one for taxes. With new Homeland Security identification requirements, however, you may need a physical address for your drivers license. South Dakota, where we are domiciled, accepts the address of our mail forwarding service for drivers licenses and vehicle registrations, but they require a utility bill as part of the identification to meet the new Homeland Security requirements. As a "work around" for full-timers, SD will accept a receipt from a motel or campground (requires at least a 1-night stay) as a physical address for identification purposes with your mail forwarding service address appearing on the licence itself. You will need to find out what is required in PA. You can probably use the address of a relative, but that address will probably appear on your license and registration.
Additional Work -
Workamper News is an excellent source for information on places that hire RVers. There are many campgrounds that offer a free site in exchange for your time. Some even offer additional pay over and above a site, but they usually require more hours.
Camper - I have never seen a motorhome with 3 bunks, but that doesn't mean they don't exist. There are some with 2 bunks, and there is a good chance you can find one of those with a sleeper sofa or a dinette that will fold down into a bed for the third child.
Good luck with your plans.