Yellowstone closes for the winter around mid-March as the snow starts to get soft, most notably on the roads which are used only by snow coaches and snowmobiles in winter. Whenever they open you can depend on snow because it is necessary to their operation in winter. There are three places to stay. Snow Lodge consists of a lodge next to the historic Old Faithful Inn, which is closed in winter, and cabins which are all within less than a half mile of the Lodge. I recommend these accomodations as there are plenty of nordic/backcountry ski opportunities and the scenery is spectacular. The snow coach drops off skiers at certain trailheads and they ski back to the Lodge. There is also an all day snow coach tour of Yellowstone which I recommend. You won't do any skiing or snowshoeing that day but you'll bareboot some places that will make memories for a lifetime.
There is a hotel at Mammoth, south of Bozeman/Livingston, MT, which is accessible by vehicle. In fact, on the north side of Yellowstone the road is maintained clear to Cook City, WY, past the northeast entrance as that is the only way into Cook City during the winter.
Some people stay in West Yellowstone and do day tours into the Park. I think the best bang for your buck is to stay at Snow Lodge.
There are some good nordic/backcountry ski routes in Jackson Hole, too. Consider a little time there in conjunction with Yellowstone.
There is plenty of wildlife up and about in Yellowstone winter. Bison are drawn to the hot springs where they can graze without digging in the snow. You'll have an excellent chance of seeing coyote and a possibility of wolves. I doubt grizzlies are up and around and are a negligible threat. Meanwhile, down in Jackson the elk congregate at the elk refuge and bighorn sheep make themselves at home there, too. Drive through the refuge and you'll probably enjoy a car wash by the sheep as they lick the road salt ... but they do a lousy job.
I can't think of a place I'd rather be in winter, aside from boating in the tropics, but then, we're quite partial to the area. If you've got your heart set on southern Utah, check out Cedar Breaks National Monument. It is of high enough elevation that snow is a bit more reliable than elsewhere. It is near Zion NP and accessed by Las Vegas.
Good to start planning early. The best destinations book up fast.