My missus's old Fabia had that. Started intermittently and got worse. Garage checked it and said that the engine temperature sensor had failed. Not a big or expensive job to replace it apparently.

IME, that's usually a lambda sensor fault, which would also fit in with most of the other issues you've described. A fault code check would confirm, it's an easy fix but usually needs a specialist tool 

code check first though

Never had to do a lambda sensor, but presumably a crows foot would also be fine? Or do you need the extra vertical grip length of a socket for those buggers?

Fucking hell. I'm glad my service garage engineers only speak German and I don't.

a crows foot might work, they are usually rusted in pretty good

Hah, what isn't?

I had the engine light come on in a rental on the 3rd day of a 1-week rental (it was a newish, low-Km car). I phoned the rental company, and they told me to ignore it.

Uh oh, I said "rental" three times...

It's autistic.