It is a dressage saddle. That's the saddle he came with, it only cost like $200 extra and it fits him really well. I can't afford a jumping saddle yet, and even if I could my dad doesn't want to get one because I can still jump in the dressage.
This is probably a big part of your problem. While you can do two point in any saddle, a dressage saddle doesn't put you in a great position for jumping. Yes, you can jump in a dressage saddle but for someone like you who has little jumping experience it isn't doing you any favors. The reason people jump in a forward flap saddle is because that allows for you to have the open hip required to follow a horse over a jump. Could you maybe borrow a saddle from someone for a little while if you aren't able to buy one?
I haven't done much jumping, as I'm sure is evident in the video. I seem to 'flop' over the jumps and bump back into the saddle. Unsure as to how to fix this though, as whenever I try I end up going into 2-point too early, which is more dangerous as Ninja tends to look at jumps before going over them and I would probably go straight over his head. Any exercises to help with that?
What I see (and what you describe) is you getting left behind at jumps. I suspect this is because the dressage saddle doesn't allow for the position that would better aid you as a beginner. And so you seem to be having trouble staying in timing with your horse. This might not be what you want to hear, but I'll give my two cents. Honestly, the best way I can think of to fix this is, at a minimum, ride in an AP or jumping saddle and maintain your two-point for a few strides before and after jumps. Of course you don't want to go over his head should he refuse, but it's going to be hard for you to really improve without good balance in two point. I agree that some horses you have to sit on when approaching jumps, like my own hothead, though usually that isn't the way you learn because it's easy to get left behind as you are riding that way. Your stirrups also look a hair long. Not sure if you'd be comfortable in that saddle shortening them, though. Gymnastics are also good for developing timing and balance.
While I have jumped small stuff in a dressage saddle as well, I don't recommend it. If lacking an AP or CC saddle, I will jump bareback over the choice of using an unideal saddle. You really can't beat a good forward flap saddle for work over fences though I get that you have to work with what you have. Maybe someone else will have other suggestions for you on possible solutions without lots of two-point, which is all I can offer.
Also when he was going over the jumps, he picked his legs up one at a time. Do you all know if this is because of lack of impulsion (my fault), approaching in trot, or just inexperience jumping? Or the jumps being quite small?
It could be all of these things. My own horse get's lazy with smaller jumps and doesn't pick up his feet. The bigger the jump is, the neater he is with his knees. Anything under about 2ft he tends to be uneven with his knees, and also when he's not got enough impulsion. Some horses are also just not tidy with their legs. It could simply be his style. Gymnastics can help teach a horse to be neat and pick up their feet.
Cross country was pretty simple, only had a couple of videos (accidentally put 2 of the same video in, sorry!). It was his first cross country outing, and I've never had a good cross country experience, so the instructor didn't want to overface either of us.
He at least looks pretty relaxed here, for it being his first x-country outing. That's a good thing.