The boards covering the mouse hole vary in size between shots and disappear completely when Tom sets up his elaborate trap to try to catch Jerry. Update: Just before the final step in the trap - hanging the cheese lure - Tom is seen with a crowbar prying at the boards covering Jerry's hole. Then the view shifts to Jerry inside the hole, then the view shifts to Tom hanging the cheese. Therefore, Tom pried the boards off before hanging the cheese.
When Tom nailed the boards into the wall to trap Jerry inside, the nails were not visible inside the wall - as they should have been with the wall being lath and plaster.
After Tom used a crowbar to remove the boards from Jerry's hole, there should have been considerable damage to the plaster - there should have been holes in the wall from the nails, and gouges in the wall from using the crowbar to pry the boards off.