I completely agree with everything Buck has said. This isn't a "Yikes!" trip, it's a "Russian Roulette" trip, where you frankly would be lucky to finish in one piece. Quite literally in your quest to "see as much as possible in the time you have" you would end up seeing next to nothing.

I will also note that in addition to the fact that you've got an impossible number of miles to complete, adding camping in October into the mix only adds to your problems. Even discounting the weather, October means you have fewer hours of daylight to work with, and trying to camp in the dark. Of course, camping also requires more time to set up and find the campground, which usually isn't right off the highway. I'll also note that if budget is your reason for camping, cabins often aren't that much cheaper than low cost hotels, although it depends on the situation and what is included with the cabin.

With 10 days, you really need to eliminate any plan that takes you to the west coast. Its just too far to do, unless all you are doing is driving straight there and straight back. Just going to Yellowstone is practically a 10 day trip all by itself, which you remember that Yellowstone is a massively huge park that requires 2-3 days just to quickly view the highlights. If you threw in a little time in the Badland/Black Hills of South Dakota on your way out, spent a day in the Tetons, and scaled back to more safe and reasonable driving days, you'd already have a very full trip for that time period.