Other mistake: Towards the end, when Riggs runs up to Murtaugh just after the car explodes, Riggs is carrying a large machine gun in his right hand. He runs up to the police barricade, but the police officer won't let him by until he shows him his ID. The police officer completely ignored the fact that Riggs was holding a large gun and could have been a criminal who wanted to use it.
Other mistake: Just after Danny Glover's shot the guy by the pool, Mel Gibson goes to handle the two girls. We hear all the exchanges between him and them offscreen, such as "don't move or I'll break your arms", etc., ending with him saying "Got 'em", and we hear a pair of handcuffs click. When we see the girls, they're either side of a tree, with the camera-side cuffs on a wrist each. But we only heard one set of cuffs (all while off-screen, so there's no reason we wouldn't have heard another set go on), and with only one set on, there's nothing to stop them running away.