Good problem solvers!

I need a good problem solver. I have a pretty good idea what might be wrong with my car, but I'd certainly like to hear from bmw community.

Signs:

Car overheates when idle(maybe after 15 minutes of idle).
Never overheates when driving or on stop signs.
When it starts to overheat I turn on heating and it cools down.
When it starts to overheat I turn on air conditioning and it cools down.
When it starts to overheat I press on the accelerator and keep it at 2500 rpm and usually cools down (not all the time, but most of the time).
No coolant leaking or coolant loss.
I didnt' do the compresion test but I bought that Napa combustion test kit that checks if there are exaust gases in coolant and it was clear (I did it few times).
I tried to stop fan with paper when it was very hot and I was able to do it without too much pressure. When I try to turn it when engine is cold there is some force stoping it but not a strong one, so it doesnt move freely on shaft, just a few rotations and then it stops.
My auxiliary fan works (I tested it with 12DC) but never turns on (even with AC on). I still think it should turn on when engine overheats (I wish I could get to that fuse #75 to check if it was blown).

My final diagnosis is Fan Clutch.
Let me hear your opinions?