Do dealerships register cars for you?

Yes, if you purchase the car from them.

Yes

This is a good question. The answer all depends on which state the car is purchased in. In some states the buyer will need to take care of the registration outside the purchase of the vehicle, while in other states the dealership pays the registration on the buyer’s behalf as it is factored in as total cost of the vehicle. In the latter instance, the dealer registers the new car at the point of sale and forwards the information onto the DMV. In this scenario, it typically it takes about three weeks for the buyer to receive their registration in the mail.