Knowing CA plate attach with the car and not the person. You can keep the plate. Cuz when you register the car in CA it's the same plate number again.

The driver license in my wife case. She maintained the CA license but have the address on the license change from CA DMV website. If you do the address change here in CT DMV they will give you a sticker for the current license. The driver license in my case I was near renewal so I kept the CA license and change me address and mailing address. Still good

If you do not register the vehicle in a certain amount of time you will eventually get in trouble when caught. You park it on the street, CT will soon get around to tax dodger who have their car in the state for more than half the year but pay vehicle tax and registration outside. That tax you owe gonna look ugly.

Here smog test is emission test. There is no plate or window sticker.