Leave camper on truck. If you want to stabilize it more so you are not rocking drop jacks to remove some weight from the truck.
As for pets I have never come across rules for leaving pets in an RV as long as they have food and are not baking. Could be a rule in hotter areas. In the vehicle yes not a good idea any place for any time.
My dog would stay in the TCer and have her water, food and toys. I would leave a vent open and a couple lower windows. Also would keep a thermometer in the camper that would record highest temps and it never got that hot. One time I was worried as I was parked in a bare parking lot. Went out for an hour came back and looked in a window and she was laying on her back fast asleep. Temps inside was 22C. Took her for a walk and she was so happy to get back into the camper that she was asleep in her spot before I left.