I don't know specifics on your system but I develop powertrain cooling systems for a living so will tell you that there are only a few things that signal the PCM to command fan: TWT (top water temp, i.e., the temp of coolant coming out of the engine heading into your radiator), TOT (trans oil temp, typically only for automatic cars), EOT (engine oil temp) and A/C head pressure.
An engine functions most efficiently around 80-90 C. The thermostat will keep all the heat in the engine (of course with some going to the heater core and in some cases a trans cooler or emissions devices) until around 80-90 C depending on the engine and application. At that point, the t-stat will open allowing coolant to flow through the radiator to maintain coolant temperature as close to intent as possible... largely based on max engine efficiency goals. The cooling fans are normally triggered as coolant temps start creeping upwards of 100 C, but are calibrated based on the OEM's goals so you could see fans switch on as the low 90 C range or as high as say 100 C.
Based on your first post I would say this may be normal for your car, but if you are not seeing the fans switch on by 101 C, I would suspect an issue likely with the cooling fan module itself or with wiring leading to it. First thing to try is to switch the A/C on and see if the fan turns on but I see that you mentioned your A/C does not work. So, I would first check to see if you have a blown fuse for the fan. If not... Given that it's an Audi, and a 2010 S5 at that... your fan module is almost certainly variable speed. Instead of the standard low and high speed, the fan module can operate from zero to max speed or pretty much anywhere in between depending on the control strategy. There's typically three wires going to it... power, ground and control signal. The latter is the thinnest wire. If you have an scope or a meter capable of reading a PWM signal, you could check to see if it is being commanded. I'm not sure if your fan module is set to default to 100% speed if the control line is pulled low (i.e., grounded) or left open circuit. A trick to trigger the fan to turn on in absence of a signal from the PCM is to ground it or open the circuit. However, my money is on the fan module being bad before the control signal being absent.
If you rule out the fuse, I suggest checking the fan module next.
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