And the Cottonwood Canyon Road....well, here's the story. The road used to be maintained by the Bureau of Land Management which is a division of our federal government, however, due to budgetary constraints ( our economy is less than thrilling these last few years...) the budget for maintaining that dirt road was eliminated. The State of Utah then had an opportunity to take over maintainance but again, due to budgetary constraints, have chosen not to, at least not beyond Grosvenor Arch headed south. As far as we know the remainder of the road is impassable.
My suggestion is that you ask locally for the conditions as things CAN change, however, I would plan on taking the highway from Bryce down to Page were I you. It is also a gorgeous drive, just not wilderness as Cottonwood Canyon is. If Grosvenor Arch and Kodachrome Basin were on your list of "must sees" and IF you have a 4 wheel drive vehicle, then I suggest visiting those sights from the Bryce side, then turn back and take the highway to Page. We used to be able to use Cottonwood Canyon as a "short cut" to Bryce as the distance is much shorter, but it always took as much time as going on the highway as you could only go 15-25 mph due to the road conditions