I don't think my dog understands the snow He's white, so I think he's just amused that it's the same colour that he is, but I have seen him trying to eat/lick it so I'm sure he understands that it's somewhat like water. I didn't purchase any dog booties for him, but I'm considering that. I know that if I walk him and it's too cold out, he will start to do this jumping thing with the paw that's too cold/hurts. He has trouble keeping warm, but will not allow me to put his sweater on him. He doesn't like to be constricted. During the winter, I try to give him short walks or simply run around in the backyard with him until I think he's tired to try to avoid walking. I know that he doesn't understand that he needs to sweater to keep warm, but I wish he would let me put it on for at least 30 minutes a day so I could give him proper exercise
I don't suggest walking your dog for 30 minutes without boots or a sweater. It won't be good for the dog. It's far too cold, and even when/if people do shovel their sidewalks, there is usually salt, and that hurts the dogs too. The salt grains get stuck in their paws and when they walk it begins to dig deeper and hurt them. This is also part of the reasoning for the jumping that my dog does.