Opening/closing interstates in the mountain states is just like in the flat ones - they try to keep them plowed and only shut things down when the snow fall is so heavy that the plows can't keep up.
Chains get used more in the mountains, especially on large vehicles. You may even be required to carry them.
Sometimes they close passes for a while for avalanche control. In Washingon (I90) that occurs a couple of times a season, and only for a few hours at a time.
It's pretty clear from the closures discussions, that the best access to Yellowstone at that time of the year is from I90 to the north. There's a modest pass between Billings and Livingstone, and a higher one west of Livingstone. More along the MT/ID border, and one in the Washington Cascades.
