Flying in close to Mesa Verde would be most likely Durango, be a bit more expensive and a rental vehicle could be very expensive especially the drop off fee. Maybe look at Denver, Las Vegas or Phoenix.

Albuquerque is closer and has enough competition for good rates.

A suggestion: Arches National Park, Monument Valley (a good ride on horseback is possible from "Wild West Trails"), Antelope Canyon (visit both the North and South Canyons), Grand Canyon, Sedona, Flagstaff, then some of Route 66 (what's left of it) to Barstow, continuing to the coast via San Luis Obispo, then take Highway 1 along the Pacific coast to San Francisco.

With three weeks I would look at Yosemite NP > quickly through Death Valley NP > Las Vegas > Zion NP > Bryce Canyon NP > Grand Canyon NP South Rim > back to Las Vegas. San Francisco > Napa Valley > Lake Tahoe all occurring before Yosemite NP.

As you rightly state, accommodation can be a challenge, and you need to settle your flights and itinerary essentially immediately. I think Monument Valley is nice for a brief visit, but very much out of the way if you're not exploring that region overall (southeast Utah, etc) - plus very hot.

We've enjoyed the Yosemite Bug (motel and hostel) a few times ... and perfectly okay when you have a vehicle - we head off before dawn each day. But you might get tent accomm in the park, which would be better.

A little confused on your route, Grand Canyon up to Bryce to Capitol Ref and reverse back down to Zion??????? That's some big double backing.
Why not do a big loop, no double backing and it's actually shorter.
https://goo.gl/maps/vEi7nhUFQHmVSSqs8

Camping or motels?

You certainly won't have any shortage of awesome night skies, all along your trip.
Kodachrome Basin State Park has some nice scenic canyons-formations, nice short hikes from camp.

If you go with the above loop, Goosenecks State Park has some very nice shots/views, similar to Horseshoe Bend.
Killer shots at the top of Moki Dugway and by Muley Point. You might enjoy Natural Bridges.
Anywhere along Hwy.12 has fantastic scenery-very photogenic, especially between Escalante and Capitol Reef.

It is very unlikely that anyone (like customs) will take an interest at the travel medicines you're carrying.
For buying medicines abroad, you must look for the molecule name not the brand name which often differs.

and arrive to GC at a decent time from bishop.

I may have missed it, being late to the party and all, but I don't recall seeing any version of the itinerary that takes you all the way from Bishop to GC in one day. That's a damn long day even without Death Valley. I guess most of it is pretty dull after Death Valley so if you wanted to drive into the night, that'd be one good stretch to it.

As for your route maps...Google maps never recommends the best route to Zion from GC, it always wants you to drive through Page and Big Water on 89. That's fine if you're stopping in the Page area, but if you just want the nicest driving route you should turn off towards Marble Canyon before getting to Page and take the more southerly route on 89A. Marble Canyon is a neat quick stop also, you can walk out on the old highway bridge several hundred feet above the Colorado River at the very start of the Grand Canyon. Rejoin the "normal" route in Kanab.

Oh and the Grand Canyon in there of course!

It's unfortunately an old story in Yosemite and other parks too. I have a copy of "Over the Edge: Death in Grand Canyon" and "Ranger Confidential: Living, Working, And Dying In The National Parks". What's left of a (illegal) BASE jumper with a failed chute isn't pretty, and I feel for rangers who have to deal with some of the carnage that they do.

Monument valley is too far off the trail. You will need to skip one of your southerly parks--I'd skip Zion.

I am inclined towards the "eastern" option myself ... Arches | Canyonlands | Natural Bridges NM | Goosenecks SP | Monument Valley | Navajo NM (if not too snowy) | Grand Canyon NP (South Rim) | Oak Creek Canyon - Sedona | Phoenix.

I would also take a sedan and stay in motels - of which there are plenty. RVs and camping to me are for more leisurely travel in warm weather - and not a high-speed whip-through in near-winter.