As QSD pointed out, when the A/C is ON, the fan should be running. If not, your low speed part of the motor is probably burnt out. You can't buy part of the fan so the motor and whole fan shroud must be replaced. If the fan is burnt, the water temperature needle should start to rise above half way, other wise the needle should stay at half or just below half all the time and yes, the fan is on the inside of the radiator, it's the only fan there is.
The compressor starting and shutting off about every 30 seconds is about right when the air is on the coldest setting.
All this should have nothing to do with your hesitation at idle (unless your running hot) and I'd hate to think you lost your fan and having trouble with the IAC (Idle Air Controller) at the same time, but you could. Everytime the compressor comes on, the idle should raise a little to compensate for the drain on the engine. The IAC does this. If not your idle could drop down to where it will shutter and shake and almost stall.
IAC position
