You will have a fantastic trip. Hope you can fit it all in, but if not, and you decide to stay longer at one stop and forgo another, you will still have a Fantastic time. The roads are all good, even the smaller highways from Durango to Monument Valley are comfortable to drive. DO NOT plan on taking the 550 (the Million Dollar Highway) over the mountains, rather head south on the 25 around Denver or before. This way you will miss the steepest climb around Eisenhower Tunnel and for sure don't drive the 550, listed on a TV show as one of the most Dangerous Roads. Durango, Mesa Verde, or Farmington are all good base locations for that area. Mesa Verde, and the Indian ruins are fascinating. Monument Valley is spiritual and lovely, camping at Goulding's is really the only game in town. At the South Rim of the Grand Canyon we have always enjoyed the Circle of Pines KOA, a large tree covered campground, with large sites, indoor pool, close to town, just really nice. Don't miss BearAzona animal park, in Williams. Fun for kids and adults alike. Personally, don't plan on more than 15 minutes at Four Corners. Lovely monument in the middle of nowhere Indian reservation, you pay to enter the area, a few stands sell Indian wears (they did have really good Indian Flat Bread being made fresh by a gal). In Las Vegas we stay at the Oasis RV park. Zion and Bryce are spectacular, but it has been years since our last trip there. The KOA in Salt Lake City is large, clean, and central to Temple Square, we are not Mormon, but the tour is interesting and the flowers and gardens are beautiful. The restaurant on the top floor is nice for lunch and has an excellent view. If you have time try to find Antelope Island. Only one small road across the Salt Lake to the Island, lots of Bison, birds, and history. Drive all the way to the end of the Island to the historical western ranch. As for your basic question, we went from Los Angeles to Yellowstone through Salt Lake City last year, we had the motorhome with toad, our friends traveling with us have a Ford light diesel pickup with a 34ft trailer. They had no trouble. I've taken these roads a few times in gas motorhomes with a tow car and never had any difficulty (with the exception of Colorado 550, which scared the heck out of me.)
Happy Trails.