My dog used to get ice crystals in between her pads and she would just stop and sit on her butt and hold her feet in the air until I came over and warmed them up. I would just cup her paw in my hand and blow into my hands a few times. That would usually melt the ice enough that I could dig it out from between her pads and then we were on our way again.
I take my dog out touring every now and then. It's a Malamute so it does really good in the snow. I try to only take her out when it's nice snow, not too deep and no crust layer. It's easy to overexert the dog I find, so be careful not to push it too far (and watch your edges! too many times have I heard about dogs needing stitches due to a careless skier).
What kind of dog do you have? My friends dog gets cold feet when we take him out touring and it's a German Shorthair Pointer. Never really bugs him to the point that he stops, but you can tell he's bothered by it. Fastest dog I've ever gone skiing with though, holy cow.
i take my dog out a fair bit and she always gets snowballs stuck in her paws but when they get too painful either she will chew the snow out or i will pull them out. you can also trim the hair on the paws so the snow has less hair to cling to. iv tried using booties but she hates them, i think your dog will survive with a little discomfort once in a while. it's kinda like getting a blister for a dog.
"there are plenty of other fish in the sea. too bad you're nowhere near the sea. you're in the desert, alone." Soulskier
"I assumed hipsters liked to stretch their assholes, since it is defiantly not a popular thing to do." cobra_commander