The inner blower fan is the one that pushes the air and not the radiator fan.
In a nutshell: The radiator fan comes on so the A/C works at its highest efficiency.
The A/C condenser (the part which sits in front of the coolant radiator) is a heat exchanger. If air isn't flowing over it, it won't exchange the heat it needs get rid of so the A/C core inside the vehicle can become cold and make you comfortable. If the fan didn't come on, you'd still be warm inside. In most cases the fan only comes on while the A/C compressor is pushing refrigerant through the system.
Does this occur always or only when it's very hot? Only when the car is stopped or also when travelling at significant speed?
The A/C system cannot make the heat vanish due to laws related to the conservation of energy, it can only move it somewhere from inside the car. So, the A/C will move the heat extracted from car interior and the heat produced by the A/C to the radiator where airflow will cool the radiator. If the car is not moving and it's very hot outside, there is no way to cool the radiator except by operating the fan.
Operating the fan is a decision with many variables, and in a well-designed system it happens when hot outside and not moving, and perhaps after a delay (the radiator doesn't get hot immediately). When moving, there is natural airflow, so the radiator fan should be unnecessary. If it's cold outside (but not so could that it would prevent using the A/C), natural convection should cool the radiator adequately.
Well the short answer is yes. The A/C requires the fan to run as part of the heat exchange process. The A/C runs on a separate circuit and is triggered by A/C system. The engine cooling system is normally triggered by the temp control sensor/sending unit or ECM/PCM depending on system, but virtually works under the same premise. Once it reaches a 200 plus or minus depending on vehicle the fan will kick on.
Many single electric fan systems normally have a high and low. A/C on triggers a high fan or hard running engine can do the same. Normally operation with A/C off fan will cycle in low as required by the engine temp spec requirements.
Others have 2 fans and one is strictly for A/C