Thank you for your service. Total of five weeks. Leaving from SC then heading to Missouri for the eclipse. After Missouri, onto Hot Springs, AR for a few days then to: Monument Valley > Sedona > Petrified Forest NP > Arches > Mesa Verde NP > then heading home. The trip is already baked regarding reservations for camping and any of the national parks that needed reservations such as Arches entry.
I agree seems like an odd trip plan, zing-zagging around. Maybe look at a map and plan a more direct route. Roads are generally good but in that section of the country expect steep grades.
Reserve Mesa Verde tours online well ahead of arrival or you will be disappointed. Arches will be very crowded and hotter than Hades. Get there as early as you can to avoid the heat and take twice as much water as you think you need. They lock the gates at Arches when the max number passes thru.
While I've not done that, it shouldn't be very difficult to do. If you look under the RV there should be an iron pipe that carries the propan from the location of the propane bottles and regulator back to very near the current propane appliances. I would tee into that with more iron pipe to get next to the location of the new refrigerator. I'd just supply it with propane in the same way as the trailer manufacturer did with what is there now. You do need to use propane rated pipe dope and such.