1) There are only two available models of horse. These are "Horse" (for elf and human only) and "Pony" (for dwarf and hobbit only). Each is available in your choice of 4 colors. These include Blond Sorrel, Bay, Chestnut, and Blood Bay. All move at the same speed and carry the same cost.
2) Yes, there are no other differences beyond color between them.
3) You can't get your horse until lvl 35. There is a four part quest line you must complete to prove that you are "worthy" of purchasing a horse.
This quest line consists of three timed delivery quests during which you must ride a horse from the stables and deliver it to three different locations before a generous timer expires and finally a timed race during which you are required to ride around a large circuit and cross through 12 gates before a not-so-generous timer expires!
These quests are a lot of fun and at lvl 35 all mobs are gray to you.
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"Well, my days of not taking you seriously are certainly coming to a middle."
~ Mal Reynolds