I have a Volkswagen w8 with a cooling problem.

Well, I think your car sounds like it is doing everything it should. If the temp is steady all the time then the fan is working properly. Also, the fan will run for a few minutes after you turn the car off because that is when an engine is its hottest.

In addition, anytime the a/c compressor is active the fan will also turn on .. The a/c compressor will be active when defrost mode is selected, not necessarily just when it's 'hot' summer time .. By design, the cooling fan operation lowers compressor head pressure .. (More pronounced the further south your zip code is).. Just random thoughts.