While the Tramp is locked into the cage, the lion is lying with his head some distance from the side wall in most shots until he gets up. But in one close-up shot just after the Tramp caught the water tray, the lion's head is right next to that wall.
When the Tramp moves his chair to set down and watch the clowns do the barbershop act, his cane falls off the back of the chair. In the next shot of him from the front, the cane is not on the chair, but in the next shot from behind, the cane is hanging on the chair again.
At 56:25 there is a small round box of make-up cream on the clothes (next to the monkey) and in the next shot it has disappeared.
After the tramp washes the shaving cream from his face, he dries himself with a towel, but the towel never touches his face. (This is probably so that it won't mess up the stage makeup.)
It is never explained why Rex went missing, allowing the Tramp to replace him for the performance.