Spyder is spot on, as far as I am concerned. A lot of the problems with your hands come from you kind of curling up around your middle and squeezing too much. YOu can see how this made you fall forward ever so slightly when the horse stopped before you were ready for him.
This is all due to him not having enough push. He is very relaxed and that's good, but he's also sucking back behind the leg and going overly slow, especially at the walk. It went so slow that for a minute there I thought "what's wrong?".
I would work on getting him to really trot out vigorously on a totally loose rein for a bit and then work on taking up the contact, encouraging him to stretch down and forward, then giving him totally zero contact and just "go!". It may be due to being ridden in a martingale that he now is 'gumby" like (as you said" and comes behind the contact.
I would work on getting him honestly forward and working toward contact with his nose well in front of the vertical.
YOU need to kill that habit of puppy paws. Was your instructor yelling "hands!" at you every 15 seconds? If I was her, I would have. (but I am really grumpy today). Your's is pretty bad and makes it IMPOSSIBLE to have smooth , consistent contact. Set yourself the personal goal of ridding yourself of this. ONce and for all.
See if you can get that lovely horse to just GO! And get some flow out of him.
Forget the test sequence for a bit.
And thank you for sharing that video. YOu have a pretty good seat and with time and work , you will do well. Really.