You are on a good start, but I fear your coach is rushing you too fast.
On the flat, you are riding off your knees. As a result of grasping with your knees, your lower leg has been shut down. It has slipped way back, causing you upper body to become unstable, tipping forward. Let go with your knee, bring your lower leg forward, and sink down into your ankles allowing them to flex and sink into the stirrup. This will make your position much more stable.
You hands have not become independent of you body, yet, and they are following your body up and down. If you had the correct contact, it would cause a hit on the horse's mouth every time you rise. As you rise up, your elbows should extend, slightly, to allow your hands to remain in the same place (relative to the horse's mouth).
Over fences Until the above is fixed and becomes natural for you, you should not go into jumping, IMO. Your upper body is tilting too much forward and is too unstable. You are resting on the horse's neck to keep from falling forward. If the horse would balk, you would come off by rolling forward.
You have a really nice beginning. You are attractive on the horse. Just don't rush things. Get more solid on the flat and your jumping will become more stable and enjoyable. You won't risk scaring yourself by feeling unstable and out of balance with your horse.
I have used a piece of twine that goes under the horse's neck. It has a loop at both ends that, when the pinky finger is slipped in the loop, it will hold the hand approx a couple of inches above the neck. The twine will hold the hands immobile. That way, you can feel how much your elbows and shoulder must flex to keep your hands at that position. When the hands stay that still naturally, you are beginning to get it.
I would do this exercise on a lunge line, so you don't have to do any steering while your fingers are in the loops.
This has helped many of my students. Hope it helps you, too.
Your leg seems to slidding when trotting, try pushing him on when trotting because hes got a sloppy slow trot.
Over fences hes slowing down at the fence push him forward and go into two point and dont push on his neck,
just lean forward bend with your help and dont slouch,