What problems should I look for? It looks like only the rear plastic panel and the back hatch that has damage, but he pivoted the whole vehicle to where the back end hit the curb and the front end swung 16 inches away from the curb. I did try driving it fro about 5 miles and the traction control lights warning lights were on. You can see the big piece of plastic that came off, and I don't think there is much left to that truck.

Did it track down the road straight? Did it pass the finger test from the tire to wheel well gap? What did your insurance say?
Lots of trash control doesn't work and obviously broke a sensor or wiring on that system in the back. But yeah just like the previous answer if it tracks okay and the axle is not Bent to the side in the back where there's uneven distance between the rear tires and the body between left and right side it should be safe to drive