I work in a hotel and I'm also from Wyoming and take frequent visits to Yellowstone, finally something I can answer!! lol
As far as hotels go, you are allowed to have her as a service animal/if you have a note. It is highly illegal for them to refuse you/her and they legally cannot charge you extra unless, when it comes time to check out, there is damage done to the room.
The pet rules in Yellowstone are very serious. The rules are in place to protect you, your pets and other people. There is potential for the dogs to attract wild animals and the animals are that, wild. I know we've taken my border collie with us but when we go hiking and can't take him, we leave in in the climate-controlled RV. (Times we were tent camping we left him home with a friend)
I would suggest calling the park rangers about the places you want to hike to see if dogs are allowed on leashes, in carriers or not allowed at all. There are a lot of fun walked and hikes that are sort of pre-mapped out so as long as you're not off wandering alone, the rangers would be able to tell you the leash/dog laws for the specific trails. There are ones with board walks as well as paved trails, then there are the dirt trails that are more like hiking.
I would suggest visiting Jenny Lake. You can take a boat ride across a beautiful lake then follow a trail that's fairly easy hiking to a place called Hidden Falls. If you continue up from there, it's a heavier hike and you reach the top, a place called Inspiration Point. The Tetons are in the view behind you and you look out over the cliffs across the lake and can see forever! Not too sure about dogs here but I'm thinking they're not allowed
Hope you have a fun trip! Yellowstone is beautiful! Sorry for the long-winded response
