The road from Gardiner to Cooke City is open year round -- except when storms roll through, of course. It's a 50 mile drive -- about an hour and a half, no stopping -- that runs through the Lamar Valley. There are restaurants and other facilities open year round in Cooke City.
I'd echo what other folks have noted. I'd be pretty careful about heading off on snow shoes. I don't know that the benefit would be worth the hassle.
You should be able to drive to the upper terrace at Mammoth, but I think the loop up there is reserved for cross country skiing in winter. The steam from the thermal pools can be both a blessing and an aggravation when it comes to getting a good shot.
I don't usually get in the Park until March, and I've been on that stretch through the Lamar Valley and not seen a thing. On the other hand, buffalo are pretty common, and the elk can be thick around Mammoth. Bighorns also hang around there. A friend of mine got some shots of wolves in that area a couple of weeks ago. The landscapes along the drive can be terrific.
My daughter spent four summers working in the Park and spent one winter at Canyon. She said that of all the animals in the Park, she was most nervous around buffalo (although in the last couple of years I think five or six people have been killed by griz in the Yellowstone ecosystem). The problem with buffalo is that they are all over the place and they like to use the roads in the winter. They can also be unpredictable. I just try to take it easy when I ease past them in the car.
Finally, if you need a break from roughing it, Chico Hot Springs is 40 miles north. The restaurant and the Hot Springs are first rate. I'm heading there in March.