Grew up in Page, rode/drove Cottonwood Canyon tons of times.
It is about 50 miles long from pavement to pavement. Road conditions vary; it is graded but can have washboards for the entire length, and/or washout sections after large rains. Do not travel if there is a chance of rain, or it has rained in the last week; the south half of the road is clay. There is a small creek crossing at the northern part, it may be dry, it may be wet, it may be flooded (see above about rain). Northern part of the road is hardpack dirt. Some blow sand along the edges or off the main road.
I've never needed 4wd on this road. Wait, I take that back, I did once when a semi blocked the road on the southern 3rd and I had to get my truck around him.
During the summer, you could easily take a small car along the road if you would like.
If you like to hammer down, it is faster to take this versus going through Kanab to almost Panguitch and back toward Tropic. But this 'time saver' requires you to kick it up to 50+ in places, which can easily be done when the road is half smooth.
I personally wouldn't pull my trailer down that road (very far anyway) unless I planned on camping somewhere along the route. As for an alternate route on your trip with a trailer/motorhome, skip it, unless it is a rental.
2005 2500 Cummins/48RE/3.73, QCLB, 4wd, BigHorn, Edge Juice w/ CTS + Turbo Timer,Transgo Shift Kit ISSPro Oil and LP pressure gauges, GDP 20/2 filters, Custom Diesel Steering Box Brace
'10 Forest River Shockwave Toy Hauler 21'
Honda EU3000I Genny