There is always some friction in the fan clutch which keeps the blade spinning along with it, even when it's "seeing" cool air. So, the fan does move some air when the engine is still cold, but very little.
The calibration built-in to the fan clutch should be such that it is intended for use with an air conditioning equipped vehicle. If the clutch is not operating at it's design level, this can account for poor A/C performance. Generally speaking, if the weather is hot, that is, ambient air temperature is fairly high, perhaps 80^ F. or more, if the engine is allowed to sit and idle with the vehicle standing, if the fan clutch is not up to par, the engine will overheat. If it does not overheat under such conditions, chances are the fan is doing adequate cooling to operate the A/C fairly well. imp