yes they do even when the car is powered down. It has to be very hot outside and the ICE has to be in the high 90 Celcius. About 97C is the threshold I see to remember reading. 100C is 212F the boiling point for water but 50/50 antifreeze and water gives a boiling point of about 224 and add pressure and you can easily get it to 240 before it boils.
are you sure its the fan or is it the pump moving water to the thermos? i heard my fan come on once and that was when we had hot weather and i was in park showing off car, the air was running. so that might have contributed to it. but that was only time and i may have heard it because i was standing outside the car
