I had the same thing happen.
Keep an eye on your Coolant Temp Gauge on the instrument cluster. If you notice the fan is running when you turn the car off, most likely the temp gauge will read zero even though your engine is clearly warm. If this is the case then you need to get a new thermostat.
The car is fine to drive, but please get it fixed as soon as you can. The fan is kept running as a safety feature due to the fact that the car [without temp data] assumes the engine needs to be cooled (as overheating engine = big problem).
If you were driving in the middle of summer in heavy traffic the fan do the same thing.
ps. The start stop not kicking in is most likely due to your battery not holding enough charge due to its age. Trust me this is a good thing as the start stop feature is a gimmick that kills starter motors and is annoying as hell. New battery = $600-$800 in these things and the reason its so expensive is due to the requirements of the Start stop feature so as long as the car starts without trouble DO NOT CHANGE THE BATTERY