I don't think it matters whether you are talking about AMHA, AMHR, Pinto, or big horses, because IN GENERAL, hunter is about form over the jumps and smoothness, while jumper is based simply on faults (knockdowns, refusals, etc). For minis, hunter is once around the course, and perhaps trotting for soundness. You want to keep the same gait - either trot or canter - between jumps and try to be very smooth. Smaller minis can do very well in this class. In jumper, you need to read the rules because after the first round, ties are broken in various ways, either by continuing until ties are broken or changing to a timed event, or maybe some variation. Also, the maximum height of jumps varies by registry.
Three is an OK age to start jumping - I wouldn't start before then. Start with ground poles, and I start them over a log or fallen tree so they know they have to jump. All of our minis love to jump (except broodmares). They need a certain amount of power from the rear end to do the really high jumps, but most minis can do the hunter-sized jumps (up to 24" usually). Heart and desire have a lot to do with it too.