Can føner be used in any other context?
No, "å føne håret" and "en hårføner" are the only words I can think of. So hairdressing only. Obviously you also have a word like "fønvind" which is a light, warm wind. Someone correct me if I am wrong!
In meteorology there's a Foehn Wind, (https://en.wikipedia.org/wiki/Foehn_wind)
Is this the same as fønvind or is that a false friend?
It looks like the same thing, even if in Norway you would not see the enormous changes in temperature described from Switzerland.
Yes, that's a warm north wind. The Germans adopted that word to describe hairdriers, and then Föhn got borrowed into other languages.