When Max, as the overweight teen, sees Leon go up in the elevator, it stops on the third floor, but when Max goes upstairs, he (as Leon) exits the staircase on the 4th floor and finds Leon's apartment.
When Max returns to the apartment to show two officers the body of the one he's "murdered," the two have their gold badges pinned to their lapels going in. On the way out, the shields are looking loose as they now appear to be the hang-around-the-neck ID badges and are not even in the same place as before.
When the thug who is beating "Patrick the Ratprick" pulls out a semi-automatic Glock pistol to shoot Patrick, you hear the sound of the gun's slide being cycled, which would put a round into the empty chamber. When Leon tells him to stop, you see the thug cycle the slide again, as if this action would "de-cock" the weapon, which it does not. In addition, no cartridge is ejected when he does this, which is what would happen had a round actually been in the chamber.