I actually know a good bit about the Falcon/Wario MU since I play Mr. Pickle a lot in tournament. Here, I think you played it well, especially in the last game, but I just wanted to say a few things:
I almost always use up-throw vs Wario because it usually leads into a free up-air string. With down throw, it'll only combo if they don't DI correctly. If they DI appropriately (down and away), then Falcon can't get a true follow-up on him and would have to depend on the tech-chase. And if you down throw them off stage and they DI correctly, then they're usually too far away for a Knee, unless you're willing to risk going for a soft Knee gimp. With up-throw, you don't have to deal with any of that. I would only down throw if I know my opponent does not know how to DI Falcon's throws.
Also, with Wario's shoulder bash, when Warios do down throw --> shoulder bash or (Insert knockdown move here) ---> forward Smash, shoulder bash covers tech in place and tech away. It completely misses tech towards them, however, because you're invincible for long enough. If you think that Wario is going to go for a shoulder bash, then tech towards them is usually the correct option, unless you are really far way from them. Then tech away would be the appropriate option. You started teching towards them near the end of the last game, which was good. Tech in place never avoids shoulder bash unless they mistime it and you get your shield up in time. But I wouldn't rely on that.
And the general idea of the MU is that you want to rack up damage on Wario through combos, strong hits, etc., navigate around his aerial spacing game, without him grabbing you and gimping you offstage. Wario has a pretty strong combo game on Falcon himself, but where he excels the most in the MU is gimping. Mr. Pickle and I usually have pretty close sets, but he almost always comes out on top because he finds a way to trick me to going towards the ledge and gimping me. And Wario's back throw and chargeable forward throw are definitely designed for that purpose. Recovery mix-ups are a must when revering against Wario.
Also, Wario has a really good crouch cancel game vs Falcon. CC down tilt can be brutal. However, I believe you can CC grab it (if you're close enough), stomp over the down tilt (which I saw you do a few times, which is good) or CC roll away. I know his jabs confirm into a grab. Common grab set up for Wario are down-tilt ---> grab, down-tilt ---> jab --> grab, or jab --> grab. CC options help avoid these set ups which, if not avoided, can potentially lead to a stock if the Wario is on point.
Also, I think the second set you posted is not Axe, but a Lucario player named Darkgun. But I can go ahead and critique what I thought about this set:
Both characters have the potential to combo each other pretty hard, but I think Falcon wins this MU handily and I think the way Falcon should play it is rather easy:
Lucario's approaches suck against characters with good dash dances. Lucario's best approach options are either dash attack or double team ---> (insert move here). Dash canceled down tilt can be good, but other than that, Lucario's approach options are extremely linear and are all beatable by abusing dash dancing and reacting to wiffed dash attacks and double team approaches. Every time you got hit in neutral vs him, it was due to him landing a dash attack on you or some move out of double team, all of which can be avoided by Falcon's long dash dance. This is why I think Falcon destroys Lucario on FD. With all that space and no platforms for Lucario to utilize, Falcon can have a field day on Lucario.
As for the combo game, I think I always up-throw Lucario at low percents because that usually leads to nair ---> up-air ---> whatever. I think down throw ---> Knee will work at higher percents, but I don't remember. I don't have a set-up at my place to test this on since I mainly play at Smashfests and tournaments now-a-days.
To end, I think I'll post some of my sets vs Mr. Pickle.
https://www.youtube.com/watch?v=kode_Mqbwgo Me vs Mr. Pickle, Winners Semis
https://www.youtube.com/watch?v=pruVp4U_Vbw Me vs Mr. Pickle, Losers Semis
I love Pickle, I really do, but I'm just about tired of always getting gimped by him lol. I think the next time I play him, I'm just going to play Marth, which is a pretty bad MU for Wario.