Regarding the fan turning on after your shut off the engine, it's a common effect of heat soak. What happens is that the heat generated by your engine while driving is constantly being pumped out by the coolant to the radiator so the engine stays at a cool let's say 91deg C. But the moment you shut off the engine, the last bit of heat in the components (block, pistons, blah blah) are still there but your water pump has stopped running. So that heat gets soaked into the coolant and raises the coolant temperature a few degrees to say 98deg C. And if your fan coolant sensor happens to be triggered then the fan turns on when the ignition is on.
I have datalogged engine temperatures during track events before, and the engine would stay under 100 deg C while on the track, but the moment the session ends and I pull into the pad dock, the temperature would spike up to as high as 115 deg C for a few minutes. It will eventually cool down of course.
This is a particular problem for older turbo engines where the turbine bearing is been cooled by the engine oil. Right after driving, the turbine can be so hot (since it's connected to the exhaust) that if you shut off the engine right away, the oil stops flowing and the turbine will heat soak itself and degrade the turbine bearing. I remember a friend of mine a few years back with a VW Corrado and everytime we go out to eat in his car, he would have to sit in the car for like 3 minutes after we get there. It's a hassle.
But in your case, I am sure your engine isn't overheating unless your temp gauge on your dash is all the way up so you should be alright.
Eddy
