I just plotted a little trip for you on my MS Streets and Trips. From Troutman to Yellowstone is 2,043 miles. You COULD do it if you began driving at 7:00 am and ended at 7:00 pm. We have driven like that before (when we were younger, in our 30's) but we were grouchy.
The trip is doable but that is a whole lot of miles. It would take you three days to get there @ about 650 miles a day. Stay in Yellowston/Grand Tetons for about nine days and then drive back for three days again at about 650 miles a day.
May I suggest that perhaps you go to Colorado instead? It would be a shorter trip and just as beautiful (IMHO) as going to Wyoming. I only say this because of the time frame you have given to us.
Just some food for thought...