I've driven bikes on -5 while snowing no problem. I've also not driven a bike when the temperature increased to positives because the bike lanes weren't cleared of the snow that fell on previous days.

Then again, raincoats are far more convenient to me than an umbrella, I prefer them when walking as well. That may make me an exception instead of a rule. As for my hands and face, can't say I care if they get wet, but my opinion on that would probably change if I was wearing makeup.