Peugeot 1007 Temperature Sensor Fault

That does sound bothersome. Since multiple systems are affected (air conditioning, screenwash, mirror heating) and they all seem to rely on temperature data, a faulty outside temperature sensor is certainly plausible. Has the sensor itself been visually inspected? Check for any obvious damage to the wiring or connector. Are you familiar with how to check these things with a multimeter?

I've had a quick look at the sensor itself, but I'm no expert with a multimeter. Where exactly should I be probing with the multimeter to check the wiring and what readings should I expect? Or would I be better off just replacing the sensor and hoping for the best?

Given your unfamiliarity with using a multimeter for diagnostics, and the fact that a defective cable or plug connection is the 'likely cause', it may be prudent to consult with a qualified workshop. It sounds like they'd be able to diagnose and resolve the underlying issue without risking further electrical problems. This should cost you about 95€.

I took it to a local garage, and you were spot on – it was a dodgy connection to the outside temperature sensor. They cleaned it up and the air con is now blasting cold air and the mirror heating works again! All in, it cost 95€ as you predicted, so cheers for the help!