Some things I noticed within the first few minutes of the first game were that you abuse wavedash smash attacks (as most low-level ICs do) and that most, if not all, of the grabs you went for were unsafe. It's interesting that you use wavedash forward smash so often, though, since most beginning ICs' go-to smash attack is down smash. Regardless, you have to stop smashing so much. When you went for grabs, they were often after Falco hit an aerial on your shield. That's the worst time to go for a grab, since any decent space animal is going to L-cancel their aerial right into a shine - which you get hit by on more than one occasion. As a side note, I did like that you were using f-air to poke through the top platform, since you've got to burn your double jump to hit through the top platform on Battlefield with up-air.
Perhaps the most important point I want to make is something that is perfectly illustrated at 2:50 in the video. You knocked him down and he was buffering shield so that he had his shield up as SOON as you were on top of him. An opponent should never feel that comfortable against an ICs. Why? Because of the grab. But he wasn't scared of it because you hadn't given him a reason to be. When you choose to smash attack, you're choosing one side of a 50/50. Smash attacks are stopped by shields and that's it (other than faster/higher priority moves). Grabs work on shields and are stopped by everything else. If you mix up your smashes and your grabs, you're likely to hit BOTH more often than if you only went for smash attacks like you do in this video. Of course, grabs are only effective if you can wobble, do chaingrabs, etc. so I'd recommend you learn those if you can't already do them consistently.
Something else I noticed is an almost total lack of desyncs. Whether you don't know how to do them right now or you're still working on incorporating them into your gameplay, desyncs are an integral part of the Ice Climber's neutral and, to a lesser extent, combo games. Even desyncs as basic as spot-dodge desync into blizzard or roll desync into blizzard are incredibly useful. A lesser-used desync that I like to use sparingly vs Falco is to jump at him with 2 blizzards when I expect him to laser. The laser will hit Popo, stopping his blizzard, but Nana will continue to blizzard, hopefully hitting the Falco and allowing Popo to grab, smash, etc. On that note, using Falco's lasers to desync your Climbers and then get mileage off the desyncs will really force the other Falco to choose his lasers more carefully, opening up many more windows for you in the now somewhat laser-free neutral.
Lastly, I'd just like to quickly say that up smash is not a reliable anti-aerial. I'd much rather see you using up-airs since they're faster and easier to hit. U-air is by far ICs best anti-aerial, although u-tilt can work wonders if you got a slightly better read.
