Some people just prefer to go barefoot! My husband for one, grew up mostly barefoot and has developed soles like leather in his very own lifetime. I certainal ly don't think that sort of thing takes any special sort of evolution!
But though he is far from immune from the cold, (hairy feet and all!) , he will still go out in the snow without any shoes. My point is that if I thought of him going of on a journey with Frodo, I would have no trouble at all seeing him go off without shoes, or perhaps strung over his shoulder.
And I believe that that was part of the trouble at Caradhras. Even a typical dog's paws would have froze up there, though perhaps not a sled dog's. I would imagine they might have tried to wrap their feet at some point, but it would not have helped. All that I can think of is that maybe a hobbit's foot hair has some sort of undercoat. As for going to battle, I see no reason why they should not have worn footwear.