RO: At what coolant temperature does the fan turn on?
Depends on the car, but usually around 105c if engine coolant, lower if turbo coolant circuit. Or if the refrigerant pressure is too high.
RO: If the engine has been turned off, the coolant is not circulating. So, what is the purpose of cooling the coolant in the radiator if the coolant is not being circulated to the other components of the system?
Hotspots. Heatsoak can and will damage the engine, particularly turbo's if you shut off the car immediately after running it hard. Coolant is definitely being circulated, most MB came with auxiliary pumps, the pump sometimes keeps running after the shutdown depending on situation.
RO: When does it time out? I ask because the ambient temperature varies based on factors like car location or season.
Location or season is irrelevant. Hot is hot. Hot, fan runs. Temp comes below a threshold, fan stops.
Time out means the timer runs out and the fan MUST shutdown to prevent battery drain, irrespective if coolant temp hasn't gone down or not. Usually 10 mins, no more.
As a result the coolant will take longer to cool in summer than it does in winter
Irrelevant. When the delta between the coolant and ambient temp is over 50C, it will cool down. What does the car care if the outside temp is 45c? Coolant temp is 95 to 100C. The delta is big enough to allow for ambient temperature exchange. Yes, it's true that in winter the coolant will cool faster due to bigger delta, but at summer time it'll still cool down just fine. Otherwise all those cars in Sahara desert or Death Valley would've gone up in a ball of fire by now.
If you put a bottle of water in the fridge so it is at 10C and then put it outside in room temp of 22C, it will warm up and come to equilibrium temp. Delta is only 12C, but that is enough. The bigger the delta and the type of material used to process the osmosis of the temperature difference will affect how quickly (or slowly) the product reaches equilibrium with ambient temp. Think hot tea in insulated travel mug vs hot tea in an all aluminum cup. Which one will stay hot for longer?
Does the time the fan runs after shutoff vary based on factors like ambient temperature?
No. Either the target coolant temp is reached after running the aux pump and fan, or fan times out. One or another.