You are not making sense. You say the engine fan does not run when the engine is running, but it starts running once you turn the engine off? That makes no sense. But we ARE dealing with German automotive electronics, so anything is possible.

1) If pin 6 of the fan module goes to zero volts with the key off, then the fan module is bad.
2) If pin 6 does not go to zero volts, that is, stays at ten or more volts, yank the connector from the fan module. If the WIRE in pin 6 is still at 12 volts, the car is telling the fan to run. If the WIRE is at zero, (with the connector unplugged) then the fan module is bad.

That is it, DO NOT make this any more complicated than my last two lines - if you do, your only choice is to take the car to a dealer, leave them a blank check, and hope they fix it (they won't, but they will charge you a LOT).

Where is the Fan MOdule?
Follow the wires from the fan motor to the Fan Module on the fender. There are four BIG wires, and two SMALL wires. The little Gray wire is communications with the ECM (make no attempt to measure what is on that wire, if you do try to measure it, please tell me you did so I know enough to NEVER help you again). The Pink/Red wire is an ignition power feed to the module, it MUST go to zero volts with the key off or the fan WILL run, this wire's voltage is the clue to tell you where the problem is.