I wouldn't describe it as a "trail" in the sense of a bike trail. It's a route that is mostly off-road. More from the map for Section 1: "Particularly in dry summers, portions of the route can become quite washboarded, making for uncomfortable riding. To prepare for it,, your bike should have at the very least a front shock absorber. A fully suspended bike is even better. The outfit of choice for most riders today is a fully suspended bike and a trailer, such as a B.O.B., in which to haul gear. In addition to providing a smoother ride, this set-up permits you to carry a greater quantity of food and water, a real advantage in some of the more remote areas traversed by the route."
Also from the map, "We discourage you from attempting to ride this route solo; in fact, a minimum group size of three is strongly recommended. If a rider is debilitated in the backountry, you will want to have at least one person to stay with the injured/sick rider, and another to go for help.
As for going north to south on the Great Divide route, the map points out that to complete the entire route at once, you would have to start in New Mexico before May 1, which would put you in the high country of CO while snow still covers most of the route. Also, the map notes that going south to north would require you to negotiate some very tough uphills. As such, AC recommends that you ride north to south.