QUESTION ABOUT A CAR TITLE AND REGISTRATION

I will greatly appreciate anyone who will be able to answer this question about a car title and registration. I live in Houston. I just bought a used car from a friend. He assigned the title over to my name in the beginning of August 2020. I haven’t gone to the DPS and officially done the transfer of the title because I’m low on money and plus I got the car after the governor’s exemption order in May. Here is 2 of my questions.

First one. The previous owner had an expired registration from 04/19. However, I got the car only this August. AM I STILL UNDER THE EXEMPTION ORDER SINCE I BOUGHT THE CAR ONLY A FEW MONTHS AGO?

Second question. On my driver’s license, I have an address in Houston that I don’t live at anymore and I have no intention or desire of coming back there for personal reasons. When I go register my car and do the title transfer, can I register my car at the address other than the one on my driver’s license?