Hi Jon, you are a world traveler, so it shouldn't make much difference what's in front of you. You'll make the best of it. But having said that, it's still nice to know what's ahead. Right? So this is what happened. I never got past window rock. I never re-connected with the Native American tour guide I was going to meet near Spider Rock. I never cycled on route 7north to the south rim. It is a tar road to Sawmill, then according to the map, a dirt road. At Window Rock, I turned around and returned to Gallup for a continuation of my trip to Chaco. The Chaco segment was another story. But anyways, this past summer I volunteered for the Nat. Park Service and at the end of my contract, I drove around the 4 corners area to look at sites that I'd like to submit my application for 2016. I stopped at Canyon de Chelly and cycled up the south rim on the beautiful tar road route 7 (heading east). I didn't find where it turned to dirt but I'm sure it's out there. Mtn bike tires should be able to handle it. I don't think touring tires will tho. There may be places with lots of sand and you'll get stuck. I think he said it was about a 7- 10 mile stretch of dirt road from Sawmill. There's nothing out there so you'll have to be self-contained.
I rode my mtn bike on some of the dirt roads out by Hovenweep in UT &CO. It's doable, I'm guessing the dirt roads out there are all the same. Obviously, bring lots of water. Canyon de Chelly is worth getting to and down into. Best of travels.