To begin, I noticed you are committing to shielding on the platform when there is a lot of instances when you don't have to. Shield dropping for the sake of it isn't what makes it useful. I would practice it more on reaction to being hit and getting down quicker altogether to mitigate the possibility of being stuck on a platform vs. a character like Marth. Also, doing it quicker with more consistency is always good, but that comes with practice. What you should work on out of a shield drop is your shield drop dj platform wavelands to keep your platform game open as well. A mixup I suggest if you are struggling with your shield drop is to light shield while you attempt to shield drop. This will make you slide off the platform if you get hit by pretty much any move, putting you back on the ground or moving you to ledge.
The first game was basically just thrown away because of those unfortunate hits out of your DJ. What I like to do first game is feel my opponent out more before I try to pressure him. Especially Marth at the low to mid levels of play, they tend to have a lot of bad habits that you need to scout before you get hit by scrubby fsmash after scrubby fsmash. That's when you download them and bait those laggy moves out. Yoshi's dashdance is great for this, and is often underutilized compared to the amsa style of tons of aerial pressure and using obvious crouching parries against telegraphed moves. You also need to use your armor more sparingly as to not get hit out of it when DJCing, rather using it to take a hit then canceling it with an aerial.
Looking at your play, you are really reminiscent of how I played late last year, Yoshi colour, tech skill and all. A top player in Canada, Rayn-Ex, told me once that I was focusing too much on my movement rather than trying to hit the opponent. This means two things:
1. Exactly what it says (ie. not getting a lot of hits)
2. You are not looking at your opponent as you play, meaning in turn that you are focusing on you own character more than what the opponent is doing
I think the same thing can apply to you, where you like to get momentum with your wavelands and create breathing room that way. This is exactly what happens at 3:52 where you die, and 3:55 when you come back, and then for the rest of the game where you are wavelanding around the stage, often unnecessarily. For one, this is bad because you are subjecting yourself to playing on the platforms against a character that has a crazy platform coverage and punish game, esp. on FoD and YS, and secondly you are giving up all of your stage control for free. Another reason why this is bad is because every time you mess up a waveland or are forced into shield and miss a shield drop, that is another 60%+ damage taken from a competent opponent. It's safe to say that a very large majority of your stocks were lost coming down from platforms or high up in the air with an aerial.
The point of looking at you opponent while you play goes hand-in-hand with not getting hit and in turn getting more value out of neutral overall. This is another habit that you have, where you try to compete with Marth's hitboxes too carelessly with your aerials. What you have to realize is that there is an opponent next to you who can easily react to your low range and highly telegraphed/high startup lag hitboxes and punish accordingly. For this, you must respect his threat zone, or the area around Marth that is the most immediate threat to you.
I HIGHLY recommend watching these two videos:
https://www.youtube.com/watch?v=RfBuAo_Bfvw
https://www.youtube.com/watch?v=gFtGQRm1bTg&feature=youtu.be
Now, Yoshi's neutral is FAR from apparent as to what you are trying to accomplish, but there are some theories. First, the general gameplan of Yoshi is usually that you are trying to force a knockdown through means of fair, uptilt, dtilt, etc. and punish hard through tech chasing and jab resets, as well as trying to gimp the opponent if the opportunity presents itself. Secondly, Yoshi wants to bait moves and punish accordingly, through his movement and his parry. Then there is the theory that Yoshi's biggest advantage over other characters is his parry, so,
"Yoshi has to abuse the situations in melee where his opponent doesn't have the luxury of messing with subtle timings.
So the game plan is essentially:
-identify where the opponent is going to need to rely on muscle memory
-direct the neutral game into one of those situations
-observe their timing
-return to that situation and get a parry"
Courtesy of Perhapsman in skype group (lol)
This is harder to explain, but this would be the next step after where you learn the basic neutral more. If you want to hear that I could PM you or make another post about it.
This is a great, semi-recent set to watch for examples of the neutral with Yoshi from someone other than aMSa:
https://www.youtube.com/watch?v=d2DELX23muo
If you look closely, you can see how leffen stays out of Marth's threat range with dashdance, and how he doesn't play the platform game exorbitantly. There are a lot of things he does wrong like his dash attacks on a CCing Marth, but his amazing neutral transfers from his Fox directly. He uses platforms a lot more on his CP to DL, because he knows Marth struggles with the platforms here and he is safer to waveland around as a baiting and neutral tool.
One last thing, you rarely - if ever - got more than a two hit string. You have to look for you best combo starters like fair and up tilt/up air when an opening presents itself, and learn when to extend for another hit and when to be disciplined. Also, ledge hop nair is really good at suffing a lot of recoveries, so work on that.
TL;DR You have very good tech skill and movement. You should work on perfecting your tech skill, learning to parry, extending your punishes, and observing the neutral and the opponent better, as well as breaking your bad habits. A lot of this will come with experience.
One last plug:
https://www.reddit.com/r/SSBM/comments/2jc3gb/nmws_guide_to_getting_good_at_melee/