RMH's are an offshoot of the Kentucky Mountain Horse. http://www.rmhorse.com/history.html
The foundation sire was a KMH, who was so mild and popular that they made a breed from him. Their comfortable gait is called an "amble". It IS comfortable but can be a little "rocky" (pardon the pun) if the RMH hasn't been strengthened through collection exercises.
I don't know much about the colors, but KMH's have a great number of what I would call "liver chestnut". The build is large barrelled, shorter legged than, for instance, a TWH. They have large hooves, which I ALWAYS appreciate, for weight-bearing, and their nose is thicker than the same sized QH. They grow very long manes and tails. I kinda think that they look like the "My Little Ponies" of the horse world.
These horses absolutely LOVE to amble FAST for long distances. If you trail ride with KMH or RMH owners on a horse that trots to keep up, you will get quite a workout.
They are generally easy going and friendly, and pretty easy keepers. I know that my KMHSA mare has a back that begs to be ridden bareback. Although there are shows for them, most owners get them for pleasure riding. I am sure that if you shop around, you'll find one suitable for your purposes. =D