Arabs and saddlebreds often have the same type of "mindset", but one isn't crazier than the other. They both can have amazing motion and do very well in saddleseat. No, they're aren't gaited - at least, none that I've ever seen in the Arabian show circuit. They aren't limited strictly to saddleseat and can do any discipline just like any other breed. They tend to favor a couple of disciplines over the others, also just like any breed does. You don't have to teach them any differently than you would any other individual horse in the discipline it is best suited. Not sure if that answered any of your questions. They're not an alien breed. Just like having an arab or a saddlebred. Yes, they'll move differently than a draft or a stock type horse, but they're very intelligent and are loyal horses.
The wind of heaven is that which blows between a horse's ears. ~Arabian Proverb