Growing raspberries in containers?

You could give it a try, but you won't get good yields. Raspberries really do need ground to grow properly. They've got a pretty good root system, they sucker (run new roots out) and hate spiralling in pots. They're very thirsty and you'd have to water and feed every day in a container. They'd give you a few berries but not enough to justify all the effort.
For pots, I'd recommend gooseberries, currants and blueberries.

Snow? We've more than we'd like. I can send some  

Raspberries can handle some shade, just not all day shade. Mine are partly shaded and give me very good yields because I manure mulch them every year and water when things get dry (unlike the UK the last year).