How awesome would it be to have an old Polonez, 125p, Trabant, Skoda, etc on streets of LA!?
Not very. 125ps are dangerous, Trabants are highly dangerous and Polonez not great either. Plus all three were very badly built and, especially Trabant and Polonez, are highly unreliable. There's a reason we have better cars nowadays. Also, in a small bump with one of those huge American cars that very low petrol prices mean they still have, you would have no chance at all. If you must get an old car with no airbags, crash frame or in fact any safety features at all, get a big one. Not a Trabant which isn't even made of metal.
Would I 100% need Polish citizenship to purchase a vehicle in Poland for the purpose of directly shipping it overseas?
No.
Next question - upon purchasing the vehicle (and registering it), can I put down a U.S. address on the registration form or do I need an address in Poland?
If you're taking it out of Poland, why would you register it there? If however you want to register a car in PL, you would need an address there.
If that's the case, can I set up something like a PO box over there?
You can, however it wouldn't be acceptable as a registered address.
my other thought is that there might be additional paperwork for the vehicles you would like to import as these might be classified as historical vehicles
Depends how old.