If I were in your shoes, I would seriously consider taking Amtrak down and back. The Southwest Chief train leaves Chicago at 3:00 PM and arrives at flagstaff just before 9:00 the following evening. I don't think you'd reasonably be able (or at least be very happy) getting there by RV in that timeframe, to mention nothing of the short time to find and buy and fix up (if needed) and get familiar with one.
To try to answer your questions regarding motorhomes, while there's some variation, a typical class A or class C gas motorhome will get maybe 7-8 mpg. I think a motorhome is definitely nicer and more convenient than a trailer for actually traveling. It's less involved to access the comforts of home while on the road, and takes a little less effort to do basic setting up and tearing down for camping. On the other hand, trailers can sometimes be somewhat more comfortable for actual camping and living, and tend to be (at least at the lower end) more affordable than motorhomes.