Mounted combat has been very bad for the game all over Skyrim. I think I've killed just about everything out in the tundra and now I'm looking elsewhere for skins. Do these deer and elk eventually come back?

what do animal skins do exactly?

I'm wondering that too..

You can sell them for (A rather small) profit, or you can use a tanning rack to turn it into leather and/or leather strips, which are used for crafting at forges.

Make leather.

I use them to make leather for armor. I've read that they respawn after three days but I've also read that they never come back. If they don't that's kind of a bummer because the tundra is quite empty now.