As a matter of fact, very cold winters tend to have little in terms of snow, at least in Poland. All that moisture freezes out of the air and there's nothing left to form snow. The coldest weather we get is usually (quite ironically) accompanied by a bright and sunny sky (the sun helps surprisingly little, though it does. Nights are when it gets truly horrible), as well as the air being extremely dry.
I get the same kind of thing here in western Washington. Our coldest days come when we have modified Arctic airmass flow in from the northeast through a gap in the mountains, something that happens maybe once or twice a winter. It's an amazing thing to experience, because the weather will suddenly shift from calm, overcast, and mild (like 40s F) to temperatures in the teens or single digits, 60 to 100mph winds, and the most crystal clear blue sky you've ever seen. Not even a hint of haze in the air. Mountains 50+ miles away seem as if they're next door. It's beautiful if you can stand the windchill.
Anyway, yeah. Temperature of an airmass is very well correlated to its precipitable water content. Cold air tends to be dry, and significant snowfall tends to happen with temperatures closer to freezing. So some places can be expected to get more snowfall as the climate warms. That said, it's still pretty complicated because one also has to consider regional changes in seasonal precipitation, which is hard. Only in the last few years has computational power advanced enough for high resolution regional climate modelling to see these kinds of changes.
I heard reports that it's getting messed up by the climate change and could even reverse its flow at some point, causing a radical shift in climate.
The Gulf Stream? That would surprise me -- my understanding was that ocean circulation takes a long time to respond to these changes, and I can't imagine how it would reverse, since the circulation is essentially governed by pole-ward heat flow and the Earth's rotation. But I could be wrong. I haven't been keeping up with the literature as well now that I don't have free access to all the journals.
Would you happen to have any good links handy? I'd love to read up on it.