From West Yellowstone - and if you want to go through Jackson, I'd advise driving south through Yellowstone and into Grand Teton Nat'l Park, then Jackson and continue south to Hoback Junction to Alpine. Then take Hwy 26 to Idaho Falls and I-15. It a scenic drive.
Hwy 22 has a very steep, curvy pass. It's doable, depending on your mountain driving experience and confidence.
x2, definitely! Have driven both these routes, though didn't tow over Teton Pass (22). Did pull our TT on 31 and it was interesting, though not especially scary. 26 is an easier route to drive (despite the town of Jackson, which you pretty much have to deal with anyway* whichever route you take) and is plenty scenic.
* = I definitely recommend taking the truck route through Jackson. Watch for the signs as you're coming into town from the north.