People haul large 5th wheels all over the country, up to Alaska for that matter. I assume that F350 is a diesel dually.
Just about all 5th wheels and Class A's are between 13' and 13' 6" high. So that is not an issue so long as you stick to highways and especially Interstates. All the overpasses are higher than that and thousands of semis (13'6" plus or minus) use them every day. Keep your speed down to 60-62 mph. Get the Mountain West directory if you are worried about it:
MWDor the Trucker's Atlas:
TA2-3 Weeks starting from the east isn't that much time. If you try for 500 mile days you won't have time to stop and look around on the way. At 62 mph you probably won't average more that 50 miles/hr after you subtract fuel, food and rest stops. So that's 10 hrs on the road. I have done 600+ miles a day starting at dawn and driving until 8PM. It will wear you out but you can cover more territory. Maybe push hard the first couple of days until you get past the Mississippi. I'd figure 4 days to Yellowstone and 4 back of course. That would leave a few days in the Badlands/Black Hills.
A week in Yellowstone will be just enough to see most of the major attractions plus a daytrip to the Tetons. Search this forum for Yellowstone there were over 2500 messages about last time I looked.
As for Walmarts or other overnight stops, you just have to look at the location and decide yourself if it is secure. I'd say most but not all Walmarts are safe. The more urban it is the less desirable IMO.