Crikey! It was cold & the wind has felt like it was direct from the Arctic, but no snow.
Week 1 training plan completed. Bounced round 10 miles today, felt really good. Loving the mad looks from the walkers - all done up in multiple layers, hats scarfs etc, and there's me in capris and a single layer still. 
Yes, you can run in snow, although last year I did find myself running with a high leg lift, to pick each foot out of the snow and then drop it back in. Disclaimer - I'm not the most co-ordinated person, I imagine most people can run quite normally in it. Ice is more of a problem than snow - easier to slip and fall.
I'm not sure I see the point of waterproof, you just end up wet with sweat rather than rain. Anyway, skin's waterproof. Getting cold is more of a problem when it rains than keeping dry, imo. I usually just add an extra shirt.