Arabian
Mustangs are used for anything any other breed of horse can be used for. They do excel at trail work and endurance though.
The stamina of an Arabian is often used in endurance races.
That's easy Quarter Horse
The Arabian horse is a general 'Jack of all trades' breed. They can do most anything, but they are best at endurance and trail riding. In the past they were also used for war horses and to improve on other breeds.
Hector, as he was written in the novel, was a draft horse - either pure or part. In the movie, he is played by a Percheron gelding.
The Palomino registry is a color registry. The horse must be the correct color to be registered with the breed. A horse can be a palomino and not a member of the breed, but the breed has only palomino colored horses in it. Palomino is a breed and a color although it is most often used as the color for example i have a Palomino Dun Appaloosa
Many breeds are used for endurance races. But the main ones include the Akhal-Teke, arabian, Mustang,Anglo-arabian, and most of the easy-gaited breeds.
Surprisingly, there is a game where you make choices to breed a hypothetical horse.
The Friesian
Andalusion
A Kladruber, which is a very rare breed of horse which was origionally used to pull royal carriages, but is now used for riding and dressage.
Freisians.