Deliberate mistake: When the terrorists shoot a man in the church and he falls on his back, both benches behind him topple (in the shot from the ceiling of the church). You can't see that the benches are connected and even if they were, the bench he doesn't actually touch wouldn't topple. (00:07:35)
Deliberate mistake: At the very end, John McClane lights the fuel trail which subsequently engulfs the plane. To do this, he tosses a lighter onto the trail which begins six inches from where he lies, having fallen from the wing while pulling the lock off the fuel dumper. Trouble is, the fuel began flooding out at the same time he fell, and the velocity would carry him some distance forward on the ground after he lands (which it visibly does); hence, it is physically impossible for him to be in the position he is in relation to the fuel. (01:50:05)
Answer: In the first movie he's a New York cop visiting his wife. In the second Die Hard, he tells the airport officer that he's LAPD and moved there because of his wife's job. In the third Die Hard film, he most likely went back to New York because of marital problems and became a New York cop again.