You have the exact problem I struggled with for ages - I like to call it "jack-knife equitation" haha! It's a combination of perching and chair seat, and it's actually not too hard to fix if you really discipline yourself. It's basically caused by a weak leg, which slides forward into an ineffective area, making you post off of your toes. Then, because it's really hard to post off of your seatbones with your legs out in front, your upper body tips forward. This ends up feeling pretty secure, but I learned the hard way that it's not at all!
First things first, you need to move your stirrup to the ball of your foot - it's impossible to have proper angles in the ankle when your foot is 'home' in the stirrup. Rotate the outer branch ahead of the inside branch; this will stop your toe from rotating outward. While standing still, have your instructor hold your leg at the spot it needs to be at. Don't let it move from that spot, and school the gaits. Chances are, your instructor will need to remind you quite often to slide your leg back, and at first it's going to feel precarious and way too far back - like you'll pitch over your horse's shoulder any second! It's just a matter of retraining your muscle memory. I have one other suggestion, and it's probably not going to surprise you at all: no stirrups! :) You can't ride in a false position without stirrups because you will come right off, so it's an awesome way to get yourself riding in a really correct position. Try to do at least 15 minutes per ride, if not the whole ride.
Figure out where your seat bones are, and get them under you, with an even pressure on each one - important, because you tend to drop to one side. Really concentrate on dropping your outside shoulder and seat bone. Then stretch your torso upward - imagine you're a tree, or Isabell Werth, or something ;) In fact, while you're riding, pretend you're a Grand Prix dressage rider, riding your Olympic mount. Sounds silly, but that visualization works SO well. I rode my 1st Level horse with a Prix St. Georges horse a few months ago and just by watching them go, I started riding him like I was riding the other horse. And HOLY CRAP, did my horse go well! It was actually quite impressive. Anyway, point is, pick a dressage rider, and pretend to be her/him.
Turn your thumbs up - elasticity needs to come from the elbow (in all gaits and movements) and not the wrist. There should never be any turning motion in the wrist, but rather a steady pull-back pressure (if needed).
Your horse is adorable, but you've got to start from behind. Right now I see a martingale and a tight rein but no hind-end engagement, which will eventually result in a false 'headset' and that will lead to curling behind the bit. Right now, this horse needs to be ridden on a light contact while you work on getting him up underneath you. When you've fixed your seat and leg you can school him to do just about anything from those aids. Only when he's fully responsive to your seat/leg aids, goes forward freely, and has mastered straightness and bend, should you work on getting him on the bit. And, indeed, it will come on its own when the correct training is there. True dressage exhibits a freely moving horse, stepping into self-carriage without a strong hand or tight rein. Like this:
I'm sure with a little blood, sweat, and tears on your part (jk haha) you guys will be doing really well in a relatively short amount of time. Good luck! :)