I rode the Divide in 2008 with a friend, after spending a little over a month touring across BC from west to east. We crossed into Montana at Roosville on August 19 and finished in Columbus NM on Oct 24.
With that late of a start, plus side trips in Montana (Glacier NP and Missoula), we had to push the pace through southern Montana, Wyoming and Colorado to make it through the high passes before bad weather and colder temps stopped us.
We spent 3 days in early October waiting out bad weather before crossing the highest pass. We got hit by a nasty thunderstorm in northern NM. But after that, getting through the desert and lower elevations of NM was easy - the weather was great and there was not much mud.
Overall we were incredibly lucky with the weather. All of September and much of October it was spectacular. The nights were quite cold though in southern CO and northern NM - mid teens a few times.
Starting mid August is too late at a pace of 50 miles/day though. Aug. 1 probably would have been ideal for our trip in 2008, but who knows what the weather will be like any other year.
I rode a touring bike, with 26 x 2.25" tires, 4 panniers, a high mount front rack and drop bars with cross levers. This is the touring setup I'm accustomed to. It worked well for me and I'd use it again. I had to get really good at using my elbows as shock absorbers though.
Most people I saw out there had a similar setup - touring bike or rigid mtb with 4 panniers. All with flat bars. I wouldn't run a low rider rack on this route, and I wouldn't consider a trailer. That's just my personal bias, but I'd hate to have an extra vehicle bouncing and dragging behind me on those rough climbs and descents.
BlueToe: a front shock could be a benefit I guess, though I've never toured with one. I guess it depends on how comfortable that bike is for you for long rides, and how well it handles with 4 panniers.
As for going it alone, I don't think a day went by when we weren't passed at least once by a vehicle of some sort. Some stretches are pretty remote, but there are hunters, ranchers, atv riders, other campers out there. Bring a Spot.