Sounds like you guys have made excellent progress! Kudos!
Does the taking off only happen under saddle, or does he act like he wants to bolt, for example, if you're leading him in a group, or leading him around the ring while other people are riding? Is it a fearful "get me out of here" kind of taking off, or a "yay, playtime with buddies, too bad for you up there in the saddle" kind of taking off?
If he's more fearful, I'd say build his confidence by slowly increasing the size of the group. Find a horse he gets along well with and ride with just the two of them in the ring. When he's confident, add another horse.
If he's more playing, be sure that you have good control alone. ReiningTrainer just posted a good thread in the Training forum on jiggy horses that may help you test and increase your control of his gaits, speed, and the strength of your "whoa." When you have good control, ride with one other horse (and a rider who's willing to work at the speed you need them to). Work on passing the other horse politely at different gaits, and when he's very comfortable with that, try having the other horse pass you, first at the same gait, and then at a faster gait. Usually if a horse is taking off out of excitement, he will when another horse passes him at a faster gait.
I'll let a more experienced jumper tackle the jumping question. Good work, and good luck!