Since the video seems to gloss over some details, I'll write this. I have no idea what your level is, so I'll just go to the absolute basics:
**** TLDR :: I don't know what 'stuttering' means, but I assume it's getting stuck in your turn around animation. Practice Dash Dancing ****
Can you consistently short hop from neutral? No? There's your problem.
Can you consistently short hop from dash? No? There's your problem.
If you're are 'okay' with those things, get used to simply dashing and short hopping and then fast falling. No lasers involved yet. Go to FD with 20xx (solo) or just plug in a controller for P2. Run forward, short hop, and then fast fall and run back a tiny amount and then repeat. The idea is to slowly move forward across the stage. If you find yourself getting stuck in the turnaround animation a lot, then your problem has to do with your SHFF timing and/or your control over the... control stick. If this is really stumping you, practice dash dancing a little. I was abysmal at it when I first started trying (I'm still not phenomenal), but you can slowly see the benefits of practicing it every single time between stocks. When I was writing this, I went back to practice this again, (something I hadn't done in a while) and was struggling with it at first. After about 2 minutes of trying I could do it consistently. It can take some getting used to.
Now the laser-y part. If you can SHFF laser in place consistently (don't even worry about laser height yet), start getting used to running forward and Laser-ing (while trying to keep momentum). One of the trickiest parts of this technique is making sure your control stick is in a neutral position before you hit 'B'. You'll know that your messing this up when you 'Side-B' during your Laser approach (or Down-B and Up-B, I suppose). It's rare, but you'll even see pros mess this up now and again, so don't feel beaten if it's hard for you.
Now, combine the techniques:
The slowed down inputs:
[1] Dash forward > [2] Short Hop > [3] Neutral Control Stick > [4] B > [5] Fast Fall > [6] Dash away > Back to [1]
As far as timing this goes, the longest gap is between [1] and [2]. If you want to fly forward as much as possible, don't let go of the control stick until the last possible moment. Laser height control is done between [3] and [4] and between [4] and [5]. Controlling this is basically frames of difference, so just see what happens when you change it up here and there.
If you are getting stuck with the transition back to [1], it's your Dash Dance timing. Often times, you'll want to move backwards a little farther, so simply Dash Dance a few times (moving slightly back) while watching your opponent. Staying in Dash Dance allows you to throw out a Laser or switch into other options like N-Air and D-Air if they get too close.
*Deep breath*
As far as Wavedashing goes, I really wish there was a magic answer here. If you can Wavedash with other characters, it's just getting used to the timing. Falco has a slightly slower WD, so the jump and trigger aren't simultaneous. Try pausing between your jump and pressing the trigger (I'm not talking by much though). If you have problems with the control stick more than the WD timing, try doing a 'Hadouken'. Start in down, and then do a quarter circle (forward or back), messing with your timing on the jump/trigger. This isn't the best option in certain situations (out of shine), but it can help when you're running forward and need to WD back.
Alright, I'm done. Hope this helps you and anybody else with this.