Nice music in the first 20s.
Well first thing I see is this. Your arm movement isn't synched with the power of your hips or legs. What I get into the habit of doing is punching a target as my leading foot is about to land, but hasn't landed yet. This puts all the weight going down due to gravity, in the punch. If your structure can handle the feedback force that is received from striking objects. You are clearly landing your foot first, then extending your arm in those movements (first 45 seconds). Well, it doesn't really matter if those movements are strikes or not, get into the habit of doing it the other way first.
I'm going to assume you're the one in the middle (camera focus), since another person has something on their left shoulder too.
When you pull one arm back, making a fist, and use a leading attack with your other arm, there are some things to fix there. One, make sure your attacking arm's shoulder is flat, nailed down in its socket, and not raised up. The shoulder must be locked straight down, while rotating, else power does not transfer and ends up jamming up in the shoulder only. Secondly, think of your other arm's fist movement as grabbing something and pulling back. To you grab something, pull it back, and hit it with your other arm. You are probably using the other arm as a sort of lever to make your other arm stronger. That's part of it, but don't forget you're pulling something real, with real weight. Easier to practice it on a body. Pull their arm to you as you punch forward. That's the proper transfer of strength. If it isn't proper, then you couldn't pull as you punch. Reference frame 1:07. I don't know if it is your gi hunching over or your shoulder lifting, but make sure to smooth your shoulders down and out with your other palm. Smooth it out while it rotates. Imagine a nail nailing it down too.
As for your punching motion, you haven't been given the correct conditioning drills for it. Here's the one i started with. Get a really large candle with a large base of liquid wax, light it, and then punch at it. Try not to punch at the base, since air can easily snuff out the base of a candle wick. Punch at the middle or top, watch the flame move. The more it moves, the more power and acceleration you have in that strike. So the point is to watch the candle flame, not snuff it out, and feel what your arm feels like when it can hit the air and make the candle flame move the maximum amount you have ever seen. Works well for range too, since people who have no range awareness will often knock the candle off (fail). Or get too far away and the flame won't move at all (fail). Close, but not too close. Great for range conditioning. Do that for a couple of days and the snap in your punching motion should get a lot better. Alternatively, do the slow movement method of taiji chuan. 2 slow movements paired with 1 medium and 1 fast, repeated over and over. But that's only assuming you know how the correct form for a strike is.
I don't really like the reaction phase, but that's not your fault. I just prefer different drills. The better weapon drill I've seen is the gauntlet. Have the testee stand with his face to the wall, so he can't see anything. Then have a line of people on both sides, with their hands behind them, then give the people on the gauntlet various weapons or tell them to attack or not attack. The testee turns around, and walks through the gauntlet, never knowing who will or won't attack once he gets past them, nor will he know until he turns around what weapons, if any, they are using. Very good adrenaline hybrid conditioning.
Also as a tip, for guns, don't stand in the bullet's vector. Move your head out of the way as you go forward and do the disarm. Do not just stand there relying on your arm strength/speed.
Your overall movements in the Reaction phase seem well drilled. You know your goal and you step in and get that goal. That's a good mentality to have, overall.
In Sparring, you like to kick with your back leg too much. It's also slow. Amazingly slow, because it's on the back leg. Almost anyone with normal training can merely step on your front kneecap or top of your foot, and you'd be trapped and hanging in the air. Learn to kick with your front leg. That has the most range, and frontal snap kicks (which I know I saw in the kata) are much more effective at covering distance than wide angular kicks from the back. Snap kicks to the groin, knee, shin, stomach, solar plexus. Can't really get any higher than solar plexus without problems.
In order to kick with your front leg, you must be able to stand on your back leg, alone, in the air, for a stable period. So you just bring your back leg up, as if taking step, shift the weight to the back leg, do a one legged stand on the back leg, and do a front kick. Exactly like the karate kid 1, almost. It's really nice if you can enter, knock someone senseless, or trap their arm, before you kick them too, as they can't run away.
Someone needs to help you with your trapping. You have two arms in front of your body in sparring, but you don't defend against the front jabs with them via deflection. Your arms are like blades. You use them to parry enemy arms so they don't reach your heart or throat. Or spleen or liver. Solar plexus too, got to protect that.
7:10 the first multiple attacker sequence. Good job taking the initiative against orange shoes there. But against the second attacker, you needed to reposition yourself. Shift yourself to the left, your left, so you end up facing your opponents in a line. This is a manipulation of angles. If you just attack people straight on, you end up surrounded. Like so.
I don't understand why people there use kicking so much, yet their speed and precision of the kicks is nowhere near dangerous. Maybe they are toning it down. But Pugpaws kicks wouldn't be so sedate like that, even when sparring. Those kicks I wouldn't even bother defending or moving away from. They have no power, so you just dive right in and get to brawling range. 7:45, the front kick, is the kind of speed that is needed. One can kick fast, but delay the ending movement to lose the impact. Without power transfer, though, people can take the hit and it won't do anything to them. Pugpaws should know more about kicking. That's more of a conditioning tool on my part.
The black shoes guy was taking it easy on you right before 9:17. However, either he decided to go full out or just got mad at something, but that kick to your helmet was real, done with real intent and precision. So if you remember that, remember it. That's useful experience. Dangerous and risky, though, which is why the coach/head started lecturing him about hitting people in the head with that kind of kick. Dangerous to hit the temple that, and even with a helmet, shock can pass straight through it unless you're using some high quality foam. Avoid high kicks, you don't have the flex and speed for it. A more practical kick is below the waist. See them try to pick your leg up there, just stomp on their arm and watch it go down.
8:45 freeze frame. That's entirely wrong. Refer to what I mentioned above about front leg landing after you made contact with striking arm. Your rear leg is up, your front leg is down on the ground, and then you hit. That transfers almost no power. You might as well not do it. And you're off balance afterwards.
For speed of punching, if you recall how fast the black helmet foot guy's punches were, the speed that I want people to obtain is at least 4 times faster than that. So if the black helmet guy's combo punches looked fast and furious to you, it's actually relatively slow and weak to people like me. The higher your arm and body acceleration, the faster your punch is. Basic physics. There's a limit to how much your arm can accelerate your hand though. Same for kicks. It's not all muscle though.
Sparring looks like a fun time you had. Keep it up. Or at least, make the most use out of it while they give it to you. To do that, though, you need to work on some fundamentals. Building blocks. Complicated blocks though.
Since your sparring seems to be some version of kickboxing and your kata looks like karate or shotokan, the true boxer's stance might help out your sparring performance. https://www.youtube.com/watch?v=AHaSHHk9Y4M
Good luck with your training and work hard. Your basic conditioning isn't bad. Your physical coordination on the kata seems memorized well. But there are some things your dojo hasn't refined for you. So you got to do it yourself.