I searched this yesterday because I just found mine does it too. Basically everything 02V8Sport says, but ToddG also said it's definentely a dealership job because its labor intensive. He said its a crushed vent and it has to be replaced. I have no idea how to even check if it's a crushed vent or clogged. Hopefully someone does.
Ah, you are the beneficiary of my wisdom, grasshopper.
First, congrats to Gus on the new vehicle. Its hard to say if the price you paid was reasonable, since so much depends on condition and mileage. Regardless, I'm sure you'll love it!
Contrary to what wi812 said, the car will chime without the key in the ignition under certain conditions. One that comes to mind is when you've left the lights on. Chimes sound when the switch is set to on or the parking lights. Chimes do not sound when the switch is off or set to Automatic. You should read the owner's manual thoroughly to understand what the car is saying to you.
On the heated seat issue, GLanga is correct, it is likely a crushed duct. It is not a wiring issue since your car does not have electrically heated seats. Instead, your car has a Peltier device that produces and moves heated or cooled air to the seat cushion or back. On the 2003 and some 2004 models, the ductwork that transports the air would become crushed after prolonged seating. In heating mode, that blockage caused the temperature to rise and activate the safety shutoff for the heater.
It is a well-known issue (my 2003 LS had it too), and is fixable. Your dealer needs to apply Technical Service Bulletin 04-18-16, which is directed to this exact problem. Basically, the fix is to remove the seat, take it apart, fix/repair the ductwork, and reassemble the seat. Did your car come with a warranty from the used car seller? If so, they should fix this since I believe it can get pricey due to the labor involved.
Here are the relevant passages from TSB 04-18-16.
LINCOLN:
2003-2004 LS
This article supersedes TSB 04-04-02 to update vehicle model years and service procedure.
ISSUE
Some 2003-2004 Lincoln LS vehicles may exhibit a condition of one or both climate controlled seats turning off shortly after initialization with one of the following combinations of DTC’s present in the climate controlled seat module (CCSM):
B2729 (Cushion Over-Temp Detected)
B2731 (Differential Temperature Fault)
B2730 (Back Over Temp Detected)
B2731 (Differential Temperature Fault)
ACTION
The conditions described may be caused by an airflow restriction due to seat cushion crush (2003 Model Year), or one of the ducts in the climate controlled seat may be disconnected (2003 or 2004 Model Years). Refer to the appropriate Service Procedure below for details.