With regard to the 'sensor' on the radiator, I think you'll find that's not a sensor as such, but a pair of thermoswitches. They simply close contacts at two different temperatures, which via a relay/relays switch the radiator fan on automatically at the speed appropriate to the temperature. So below maybe 95(?)°C neither switch will be closed and the fan will not operate, above this (guessed) temperature one of the switches will 'make' and trigger the fan to come on at low speed. If the temperature still doesn't come down, the other switch will close at maybe 100°C and bring the rad fan up to full speed.
Should be able to test it by getting the car up to the temperature where you'd expect the fan to be on, unplug the connector from it, then measuring for continuity between the pin the brown wire was connected to and each of the others.
The wiring diagram is a bit complex in this area so I can't guarantee that there isn't more to it.
The sensor that tells the ECU the engine temperature is the one that gives the fault codes you mention if bust, that's at the rear of the head of the engine. Don't think that one can switch the rad fan on, or fail to if broken.
Edit: If the fan runs rough, as you say, it may well have blown a fuse due to pulling too much current. This seems more likely than a failed radiator thermoswitch TBH.
Don't think the overheating will be causing any of the other fault codes, but the faults that are causing those codes may affect engine temps, maybe.