Looks like a fantastic trip. Are you going alone or with others? I finished a long road trip with a good friend back in September. Most of our nights were in campgrounds, and despite the time spent hiking, traveling, etc. the times we spent in campgrounds were some of the most memorable; so if you are going with others I would consider more camping! However, one thing to keep in mind, is that almost all campgrounds are shut down in the winter -- as in, gates are locked and access is absolutely forbidden. When I previously did weekend road trips to campgrounds in the winter, I purposely drove through plenty of public land to make sure I could find a campground with an unlocked gate, or, as a last resort, park the car at a turnout somewhere and pack in my camping gear, hike out 100+ yards from the road, and legally camp.
I will recommend one leg of your trip that I am very familiar with -- SLC to Jackson. Instead of your Google route, take I-15 up to Brigham City and take US-89 all the way up to Jackson. You will go through Logan Canyon, past Bear Lake, go through some very nice mountain valleys in Wyoming, and take a nice ride along the Snake River. The ride is much more scenic than the Evanston-Kemmerer route, and it takes just about the same amount of time too -- not to mention that wind-blown snow is also known to shut down some major roads along your planned route.
Finally, when driving from Jackson to Kentucky, it is worth your while to make your way through the Rapid City area to head towards Mt. Rushmore, the Black Hills, and the Badlands. The roads SHOULD be open and you can enjoy this area in less than a day while making a direct route through. I made a similar route when moving from SLC to NC last November, and I really wish I had more time to spend there in South Dakota.
Enjoy, and drive safely.