Heck you can do so much with miniature horses. Small children have no problem riding them as long as it is done appropriately.
They really excell in driving, jumping, liberty, halter obstacle etc.
To train one to jump start with poles laying on the ground. Start by letting the horse look at the poles and smell them.
Have the horse walk thru the poles slowly. Keep repeating until they learn to pick their hooves up and not touch the pole.
Gradually start raising the poles off the ground.
Always align your shoulder with the horses head when the horse goes to jump and hold your rein hand up. Many mistakenly get way out in front of the horse which will cause it to overjump. Or they lag too far behind which will make a horse balk. You have to have that forward mometum and energy going.
Here is my son jumping a mini. Notice the body alignment
You can do a bunch with minis!
If they are big enough you can break em to ride.
They are great at driving, showmanship, in-hand trail, jumping, liberty, and if they have good conformation: halter classes can be fun (especially color classes)
We own quite a few miniature horses and shetlands, and they're great! We go for trail walks, do obstacle courses (teeter totters, barrels, cones, poles, tires, jumps, in and outs...pretty much anything goes!) go for therapy visits, play 'dress up' attend showmanship and halter shows, show off in parades, lunge, and I'm teaching my three year old mare to drive! Both of my minis swear that they can't jump though ;) silly horses.