I wish I'd seen this earlier. I knew as soon as I saw the floormats that you would have problems.
That car is a lease return from a rental company, specifically Enterprise(their logo is on the mats). You can get a good deal on that kind of situation, but you have to be EXTREMELY careful not to get burned. The fact that the rental company left the mats in the car bodes ill, because during the normal return process, the car would have gotten new, factroy floormats. Which means that the car didn't go through the normal return process, which can only mean that it was returned with a problem and rejected, or that it was bought from lease, and then was sold as used rather than preowned.
Either way, somethings up with that car...
LSes are nice cars, but they're still Fords and have all the build and wiring problems that go along with that. Get an SSEi for the money.