Welcome!
I am gonna try and help out as good as I can. I will use some words that you might not have heard of before - if you got questions regarding any of the stuff I say feel free to ask.
Falco's laser is a game within melee. It will take alot of experience to be able to handle it properly.
There are a few things you can do to work around it and some of these are harder than others.
The best thing you can do is to Power Shield the laser thus turning the situation around literally.
Power Shielding is really hard - but of course becomes easier with practice.
There is a neat trick to power shielding easier tho. Actually 2 tricks.
1. Hold forward before you shield. If you angle your shield forward you are more likely to land the power shield.
2. Crouch and shield out of crouch (you will stand up and shield in 1 frame). If you shield when the laser is above your head, you will power shield it guaranteed.
In order for you to try and go for power shielding it is very important that you can wave dash reliably. The reason is that you will need to Wavedash OoS (Out of Shield) after most of all attempts at a power shield. If you don't you will put yourself in a disadvantage every time you try. Shield Release (simply letting go of shield) hinders your mobility by alot.
Besides Power Shield + WD OoS,
You can
-Shield the laser and WD OoS.
Shield it and choose an aerial option such as SH Needle, SH Fair, FJ Needle.
-Crouch under almost all heights of lasers.
If you get hit crouching you will CC the laser, thus mitigate alot of the hitstun.
So learn how to act really quickly out of Crouch. And also out of the hitstun of a CC'd laser.
You can actually SH over most lasers. (People bash on Sheik's SH for being too high, but in this situation the height is actually really good).
And then the somewhat obvious option is to move up and down from platforms.
It's actually a legit strategy to Shield a laser and then Full Jump and Waveland on a sideplatform. Or simple jump as soon as you see the laser coming and avoid it completely.
The important part here is to know there are alot of options to how to deal with the laser - And utilizing as many options as you can.
So if the opponent shoots a laser and knows what your answer to that is (SH over it every time) - that is bad. But if you can utilize just a few of the options I mentioned, you can have a mixup as to how to answer them and stay unpredictable.
There is also another aspect of lasers, that is really important, and that is to note from what distance the Falco is shooting from.
If he shoots too close to you, in neutral - you can hit him, with certain moves, before the laser comes out.
Ftilt, Dtilt, Dsmash, Jab and Rising Nair comes to mind.
I hope that helps you.
Oh and btw. I am not expert on how to handle Falco's laser. But these are the things I know so far.
----
Ok about chain grabbing.
You cannot true chain grab fast fallers before high percentages, but you can grab them over and over using what we call the Down Throw Techchase.
I recommend watching this video because it explains the basic mechanics of it
https://www.youtube.com/watch?v=OkMsQGlcrgU
Then you can watch this Sheik specific video
https://www.youtube.com/watch?v=o7l1IWUPznM
---
About the Nair. It's really complicated and an extremely broad subject.
I'd love for you to ask more specifically about what troubles you about it.
You cannot simply counter Fox's Nair with a simple answer that works every time
