I bought a used coupe last weekend and at the dealer they charged me the sales tax and government fee required to transfer my plate from my old car (trade-in) to my new car. I have a few questions:

Do I still need to file papers with my DMV for the tag transfer and registration? When they ask for money do I just show them the invoice showing I've paid it already? What if the amounts are different?

They charged me more sales tax than they should have, and said that they were doing it to cover their own rear in case the taxman came down on them (and they underpaid), so how do I get the difference back from the state?

Thanks!