As others have said, a rig that size is not uncommon and is used all the time for fulltimers. Your TV can handle that FW. As for towing in the mountains, not a problem for your setup, but you are another question as that depends on your experience and capability, which you most likely have. As for traveling that distance being hard on the rig. No. But definitely make sure your tires are aired up correctly. Cold.
We have a Trucker's Guide to tell us the roads to travel and be wary of, overpasses, etc.
400 - 500 miles a day is a lot. We travel about 300, then stop. Even though our setup travels smoothly, it's still taxing and you need to rest. You've been given advice about Cracker Barrel parking lots. You can use Google Earth or Maps to check out parking areas in advance.
Take the generator because you never know when you will need it. One trip we took to Canada, we planned all full hookups. When we got to one of the Provincial Parks, the full hookup sites (no reservations - we took our chances, hence make a reservation, just in case) were taken so we took a dry site. No generator. Not a problem but it would've been much more comfortable to have the generator.
As for CG's fitting your rig, or your rig fitting in the CG's - not many we found in Yellowstone will fit big rigs unless you are VERY comfortable with tight CG's. We've either stayed outside the park in West Yellowstone at Grizzly, or 2 miles north at Baker's Hole, a FS CG with electricity at the sites and big rig friendly, or stayed at Colter Bay RV Park between Jackson and the Grand Tetons. One year we stayed off the Togwotee Pass between Moose Junction in the Tetons and DuBois Wyoming at Falls CG. It has an RV loop with electricity and is big rig friendly. Both Baker's Hole and Falls as a water faucet at the entrance to fill up before you pull in. Baker's Hole and Falls are first come, first served, but in September we've been ok. Baker's Hole is more popular since it's closer, so what we've done is to stay at Grizzly. Go over to Baker's Hole in the morning, find a site, put our name on it, pay for it, then back to Grizzly to get our rig.
Weather in September is great. That's when we prefer to go. Crowds are down but there are still plenty of people enough that some of the CG are still full, hence the technique previously mentioned on getting a spot at Baker's Hole. As for the time span you have, not sure that's enough time to see all without a lot of long days of driving. We took two weeks from Boise Idaho to Mt. Rushmore and only stopped overnight at Devil's Tower and the Little Bighorn Battlefield. What you're wanting to see is much, much more. But, it's doable if you overnight in the Mt. Rushmore area, drive through the Badlands, and save the rest of the time for Yellowstone and the Grand Tetons.
Hope this helps.