More bacon than the pan can handle
Welcome to the RTA Forum!
Let me start by saying that if your expectations are to drive 15-20 hours at a time, then you absolutely have a major problem with long drives. Doing that in a single shot is dangerous enough, but trying to do it as part of a long distance trip, where you need to be on the road for multiple days at a time is down right homicidal. By day two, you'll be every bit as dangerous as a drunk driver, a problem that would only get worse as you continue moving forward. Having multiple drivers doesn't change the reality of fatigue, and even if you manged to not kill someone else with your car, I assure you, the 4 of you will be more than ready to kill each other from the combination of fatigue and close quarters. 600 miles a day should be the upper limit of what you try to cover in a day, from both a safety and enjoyment perspective.
The reality is with 2 weeks, you don't have enough time to hit all the places on your list - even if you were just going one way (and based on stating you want to do this as a loop, I assume you need to return to VA). If you were at a full 3 weeks, you could theoretically drive to all those places, but even there, that's all you would be doing, is driving from place to place, without time to stop and see anything.
Unless you can get significantly more time, I'd strongly recommend cutting your list in half - either do the southern things, like Grand Canyon, Los Angeles, and Arches - or the northern locations like Banff, Glacier, and Yosemite. Frankly, even there, trying to fit either of those in those with less than 3 weeks would be a pretty rushed trip, but at least it would within the realm of reasonable.
The price you pay for free camping.
As mentioned above, fatigue is a huge issue with long distance drives. Unfortunately it sneaks up on you without your being aware of it. By the time you become aware, you have already been driving for hours in a fatigued state. You may feel OK, and not tired, but then I have never heard drunk driver say/admit that he was not fit to drive. Worse still on a multi day trip fatigue is cumulative, each day adding to the previous. [I recently took a day trip locally, and there were numerous roadside billboards telling me - and all other drivers - that only sleep cures fatigue.]
Furthermore, a recent email which came into my inbox was headed the 20 best free camp spots in the US. The bit that struck me in particular is that it emphasized that roads in (and out) were not the sort of roads you want to drive in the dark or when you are tired, they were usually long and slow. No reason to think that most other free sites are any different. It is the nature of those sites that they are mostly isolated. The price you pay for free camping Neither are these spots ever near the highways or an urban area, nor a tourist attraction, such as the parks.
In other words, you'd need to arrive in daylight, leave in daylight and drive a fair distance to the attractions. Beter off camping inside the parks, at least you can enjoy it from he moment you wake to the moment you go to sleep. If that is beyond your budget, you may have to delay this trip till funds are available to enjoy all of it SAFELY!
Lifey