Did the same route last fall...
US-16 is the better route across the Big Horn Mts (vs US-14) but still a decent climb and drop. That's really the mountainous part of getting from Devils Tower to Fishing Bridge.
At 10 Sleep you are past the worst, the route isn't bad at all (north to Greybull then west to Cody). No reason to go way south to Thermopolis.
As far as tires, assuming there are no other issues (balding, bubbles on the sidewall, etc...), 3yrs is fine. Typically around 6-7yrs is when you start to consider tires having aged out.