Lmk with the buzzer I was thinking you might have somehow enacted the immobilizer. I don't know anyone who actually has yet nbut it cuts the power to the starter. At least that's the information I found about the aveo immobilizer. I'm still not buying we have an actual immobilizer.

From the sounds of it you your tpms light is on because you are still at the wrong pressure. Too low or too high it will be on. Or if you don't have the factory wheels, no sensors, one bad sensor or so
Many other issues with them. It's a waste but federal government knows best. The pressure has to be within 1-2 lbs either way of the sticker in the door.

The short to ground fel of the starting circuit sounds like the starter relay or the starter itself though. But that's hard to say as I don't have the car. But you are not experiencing a common problem. If that is what you are looking for.