Hmmm...your most debilitating problem was your obvious nervousness, but I ultimately don't think that will be too hard to get over. Let's see what else I should pick on...
Firstoff, I wouldn't use the Fireball, especially not vs Captain Falcon unless I'm ledgeguarding or wavedashing backwards towards a ledge. The Fireball is really laggy and for what it gives you, and it is really easy to avoid and short ranged. You got punished a number of times for trying to use Fireballs against Falcon, and the thing about Falcon is he is all about punishing your openings and does it really well against a laggy and easy to avoid move like the Fireball.
Next, learn to use spotdodges. I just don't remember you using one ever. It probably won't work too well against an N-air, but it will work against a D-air or a Knee.
Another thing that I think is worthwhile to try is comboing out of a single Jab. As a Mario main, I can vouch that D-smashing from a Jab pwns. Luigi might not only be able to do that, but if you can anticipate your opponent, you might be able to wavedash -> Jab -> Up-B, which is also effective. It's something I'd keep in mind and try to work on.
I'm not sure if it is just me, but I think Luigi should try to stay grounded more often when facing against Falcon. Falcon owns the **** out of Luigi in aerial mobility since Luigi's aerial mobility sucks. However your ground mobility easily rivals Falcon's, and might be better. If you must jump, short hop, and try to use the B-air a bit more since it has longer range than Luigi's other aerial attacks. Being above Falcon is dangerous considering how his U-air has particularly huge range. I'm not particularly sure how to tell you to get around the N-air except that you should kinda anticipate around it and that Fireballs don't work on it.
Grab combos own Falcon. Practice those a bit. I think you might need to grab a little earlier. The hitbox for the grab doesn't come out immediately, so if you are pressing Z right where you think Falcon should be grabbed, then you'll probably miss him. You need to press it when he's a little above your head.
Eh, just my two cents. I hope it helps.