Not an answer, just a brag! Went to Lapland last week and we saw a herd of wild reindeer come out of the woods while waiting for a bus (us, not the reindeer!).
Also bought some slippers and a hat made of reindeer skin - they are really warm, so if their skin is that warm then they will have no problems in the cold.
They have also hollow hairs which traps air so insulates them against the cold. Their feet have also adapted to broad cleft hoofs to cope with the slippery conditions they mostly live in.