Other mistake: When Jay opens the post office sorting machine and we see the multi-armed alien, there is a stereo playing rock music quite loudly. There's also a hole on the other side of the machine (to fire the letters out of) - shouldn't we have been able to hear the music through that hole before Jay opened the machine?
Other mistake: When Agent J. is riding the Giant Worm in the subway, you see him getting hit by a metal bar on the top of the tunnel smack right in the head at amazing speed, twice! Yes, twice that happened. And later, when he is sitting on the bench with his partner, you see his head looks completely untouched, no cuts, blood, scars, absolutely nothing. Certainly he would at least have a little mark, but nope, nothing but a little sweat. Does Agent J. have a hard head or what?