I'm not much of a jumper, so I'll stick to critiquing the flatwork in your first video here:
Your position here is not bad at all. I agree with what tinyliny has said, so won't elaborate on your seat too much. I think your seat will come further along with more practice.
There are moments you tend to fall forward on your pelvis (10-13 second) and you can see that your back arches more (need more core to keep yourself centered here - think blowing out a candle/ belt buckle forward) and your horse bounces you out of the saddle a bit. However, I think this may have also occurred because how much inside leg you are trying to use, which is causing tension in your seat and back at times.
Your leg, at several times, is rotated outward at the thigh and lower leg. Near the end half of the video, your thigh is a bit better and has more contact with the saddle, but I can see your horse doesn't move off that inside leg very well. This is unproductive to your sitting trot due to the tension it causes. I agree with ClearDonkey and would reduce sitting trot and do more posting trot for the above and also for other reasons I will state below.
Lastly, I agree your quite gentle with your hands, but I do see you using your inside hand as a 'crutch' here and there (0:40 - 0:47) without your horse really responding to that inside leg. What this can cause is your horse to weigh in on their inside shoulder. You'll see below that this is something your horse already has a habit of doing (not saying that it is your cause, just that it may not be helping). Your horse is giving but not really providing true bend. To check for true bend, give your inside hand forward and see if your horse will hold the position. If not, then he really isn't responding to your inside leg.
I like what you did at 0:04 in lifting your inside hand gently to assist your inside leg in standing the horse up. This trick is handy for horses that persist in leaning in and while it does not give immediate results, it does help back up your leg in a way that makes a horse respond. As you can see at 0:05, this did appear to help and then at 0:06, your horse dropped his inside shoulder again and that is why his head came up. It then appears that you used your inside leg to bring him into position again (to which he relaxes), then persists in pushing the inside shoulder in (0:08). All this is really saying is your horse is dropping his shoulder periodically and you are bringing him back together, but not really training him to respond well off the leg.
0:21 - 0:24 - You are having some good moments here and your horse starts bending and your inside hand is slightly up. At the end of 0:24, your horse pushes in again and at 0:25-0:26, you use your inside leg/hand lift to push him over. He does a little, you give with your hand and he relaxes BUT he did not really respond to your leg quite enough because at 0:27 that shoulder drops in again and so repeats the pattern.
I'm sure you can see a pattern from above, that you need to get your horse responding off leg quicker. You are working too hard with your leg and need him to start holding his side of the load. To do this, you need a system: you suggest, you ask, then you tell. If you are to the point that you are starting to ask him to move over with your leg 3x and he doesn't or he only gives the littlest bit of effort, then you need him to give you a BIG response, almost an exaggerated movement. The next time you ask, he will be quicker to respond. Your goal is to have him bend and/or move over off the smallest possible signal (seat eventually). For example, you are asking him to stand up on the inside and you have asked with your leg enough times. Now, I'd be asking him to move sideways on the circle with energy. In general, don't worry about disturbing the frame, don't worry about him moving too sideways in training, as these ultimately make it so that it is easier to manipulate his body into alignment where he can easily relax over his back. Thus, your aim should be to set up his body correctly first before you worry about the frame at all.
Secondly, I agree he needs to be more forward, which is why more rising trot would be best for now, but if you are doing more sitting trot for yourself, then practice away :) He just is not strong enough over his back to consistently keep lifted.
Your horse's left side is his stiff side, meaning that your horse is actually tighter length wise on the right than he is on the left. So, you need exercises that work on lengthening the right side such as spiraling in and 'leg yielding' out, but with a bend. Since your horse is familiar with travers (haunches in), that would also be a good exercise to work with on the left rein. However, I would work on it at the walk, so that it will be easier to correct him from falling in. I usually work on travers at the walk instead of the trot until it is easy for me to ride and fluid for them to do. I'd also ride the left in a bit of a shoulder fore to help set him up for bend.