Revealing mistake: In the scene at the motel where Chigurh kills three men, he goes to sit on the bed where he removes his socks. In the background one of the 'dead' men opens his eyes and looks around.
Revealing mistake: Chigurh's shotgun has a black finish and a synthetic stock. While these are popular on today's "tactical" shotguns they were not available in 1980.
Revealing mistake: When Chigurh is preparing to blow up the car in front of the drugstore, he uses a piece of cloth as the fuse. He removes the gas cap, places a piece of cardboard over the opening, and lays the cloth on the cardboard. He then lights the cloth and walks into the drugstore. After the flames burn through the cardboard and reach the fumes from the gas tank, the car explodes. If you watch the burning cloth as he walks into the drugstore, you will notice that it never moves, not even when the car explodes. If the explosion had been real, the burning cloth would have been blown away from the car by the hot gases expanding out of the gas tank opening.
Answer: "Blood Trail (end titles)" By Carter Burwell, the Coen's frequent composer. The Film does not have a soundtrack, but you can listen to it in Carter Burwell's website: http://www.carterburwell.com/projects/NCFOM.html.