neck reining/gateing

I've neve done it myself but it doesn't sound terribly hard.  One of the more recent issues of Horse Illustrated has instructions...I'll check on which issue it is.
 
For the bouncing...work on your seat.  Some horses ride like jackhammers, but having a strong seat will help you stay in the saddle.
Does your horse gait while in the pasture?

no he doesn't gate at all.

Neck reining is very easy to teach, especially if they know how to plow rein/direct rein.  I teach my horses to neck rein by first asking for a turn by laying a rein across their neck like I was neck reining.  Then a couple seconds after this cue, I then follow up with the corresponding direct/plow rein.  As long as you consistently do this, they will start listening to the neck-rein cue before you ever have to ask with the direct rein.  They learn it even faster if you use leg aids as well.

The bounciness may be due to his conformation, however its more likely due to becoming uncollected and strung-out, especially if he is smooth initially.  This will take time to build collection and get him working the back end.  Sometimes I find working circles and various speeds and serpentines helps prevent getting too strung out, but I am sure there will be others who will be by shortly with better advice on getting him collected. 

Gaiting - well, just because he is from a gaited breed, does not mean he will gait.  Does he gait out in the field at all without a rider on his back?

no, i have never seen him gate in the field, but its always possible that he does when we arn't watching

I don't work with my filly enough but one thing I learned from a trainer is, when you're doing ground work, you take the lead rope and run it along the neck about the location the rein would be if you were riding the horse.   With one hand (the aft hand), hold the rein and with the hand closest to the head, using your fingers along the "rein line" put pressure on that line just until he moves away from the pressure.   (then you wipe away the feeling like rubbing, according to this trainer)  So you apply pressure, the horse gives, you let up immediately, and wipe. After the horse catches on you might have them move further.  Only do it a few times each side for each session.
 
On gaiting, if he can get into it and then lose it, you need to settle your seat, squeeze with your thighs, tighten your core, sit deep, lean back a little, lift your hands and bring your horse back.  You may have to walk and start over a couple of times.  Get the gait, try to hold it.  Just over a walk, just under a trot.  Sounds like someone has been speed racking your horse.  Just a guess.  That feels like a freight train to me! But it's cool.  But it's work to gait.  Just go slow.  My Standardbred can do it.