The biggest problem health wise with Morgans is that they tend to be, as a breed, predisposed to insulin resistance and chronic laminitis. Many get chubby rather easily.
I hear some people complain that they can be on the hot side, but I've also handled some very laid back, almost lazy Morgans as well. Its an individual thing. There are also a couple of types within the Morgan breed. The newer type, I think, are the ones that tend to be a little hotter.
Discipline-wise, there isn't much that Morgans can't do fairly competitively. Hunters, Jumpers, Saddleseat, Dressage, WP, Reining, Barrel Racing, Driving - you name it, there's been a Morgan doing it, and probably pretty well. They're originally bred to be all-arounders. I do believe that there's some historical precedent for Morgans as preferred cavalry mounts.
Purchase expense is very individual - like any breed its going to depend on the individual's conformation, training, potential, and the geographical market conditions. Ongoing maintenance expenses are also very individual.
I personally like Morgans. I've owned a Morgan cross, and ridden several other Morgans, trained for a variety of disciplines, and never been disappointed.
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown