Thanks! Don't worry about the questions, I was actually trying to encourage them when I asked for thoughts. Been awhile since I looked at your moveset but I'll do so again.
This ended up being a long post because I explained so many extra details so hopefully you don't mind.
I'll use + to show my talking points and if it doesn't relate to what you want to know then thats up to you as to whether you want to read it or not
+ Story Behind Choosing the Animation
Oh, yeah, my idea with Court Change was kinda outside the box and could have been explained a little better (but I thought I'd spare the details seeing as I'd written so much anyway) but the explanation in-game for the move is so vague that I decided to get a little creative with it. The dex explains it as a "mysterious power" and I figured, "What is the main mysterious power in Sword/Shield? Well that would be the energy that comes from Eternatus of course!" so the idea was that Cinderace is tapping into this power that is generally used for dynamaxing and as a result the excess energy (signified by black smoke and red light) is coming out of him as he uses it.
+ Cinderace's Court Change Animation
To answer your main question, Cinderace is acting out in frustration like when a soccer/futball player might rant wildly about a foul or injustice on the field that isn't being called. The idea is that Cinderace would be angrily raising and shaking one of his fists up (in front of chest/face) as he says something like "cin-CIN-derrrr-CIN!" as the eyes and eyebrows show his anger and 'storm' clouds puff out around his head (kinda like when steam is shown in animation to show boiling anger). During all that, the same black smoke infused with red light is forming an arcing barrier in front of Cinderace with the red energy specifically stretching out a little (of which looks similar to Mii-Swordfighter's reflector animation).
+ "Would Pyro Ball be able to bounce of the stages of the screen?"
With Pyro Ball, if there is no physical wall then it can't bounce. So if Cinderace is by the ledge and shooting it offstage towards the blast-zone on Final Destination (for example) then it won't come back.
+ Lists of Moves I'd Keep
I'm trying to figure out exactly what you mean by your last question but I'll answer it in many ways so that hopefully I answer it and you know my thought process.
If I had to limit myself and could only pick 4 moves to represent Cinderace in a general sense then they would be; Pyro Ball, Court Change, Bounce, and Headbutt (or any of the moves associated with the word).
If what you meant by was the specials specifically then I'd say; Pyro Ball, Court Change, Bounce, and Feint.
If you meant out of all the moves in the set which are most important to me that he keeps then I'd say; Pyro Ball, Court Change, Iron Head (Fair), and Flame Charge (Dash attack)
+ Explanation for the Recovery and how it relates to Bounce:
I ended up picking High Jump Kick for the recovery because I noticed it was a recommended move on some sets used so I figured it'd be good to also somewhat represent the competitive scene of Cinderace and the move fits overall very well with the theme of the Pokemon (think Rabbit/Bounce). I chose it to be the reverse of Incineroar's recovery because in the dex explanation of the move it says, "The target is attacked with a knee kick from a jump." but I still wanted Cinderace to use its legs when recovering and I had no intention of giving Cinderace a great recovery so I added the second part. I probably could have spoken more on the fact Cinderace doesn't have a good recovery in this moveset but the thought hadn't occurred to me and I think its good for balancing this set, and again, fits with the theme of the Pokemon.
Originally I actually had an idea to do a variation of Pyro Ball but Cinderace would shoot it vertically below him, using the momentum of kicking it to
bounce the desired direction. While I still could have gone with it, High Jump Kick just seemed like a more simple and balanced route to take. You could also interpret the move as a type of 'bounce' seeing as Cinderace jumps in the move.
+ I Don't Agree with Limiting things to 4 Moves and Here's Why:
The way I see it, Pokemon can use more than 4 moves. In the mainline games limiting it to that small number is just a way of keeping balance within that game genre.
Although in the anime they try to keep it close to the lore of the mainline games, they haven't always upheld this and the show is made specifically for younger kids with the idea of advertising the games. With all of that in mind, it makes sense to stay as close to the game mechanics as possible, only veering from that in order to liven up the plot. One example of where more than 4 moves were used is in one of Snorlax's battles when Ash was using him. I'd suggest looking it up for more details if you aren't familiar with it. Another thing about the anime and general advertising for Pokemon is that the vast majority of the creatures make audible noises which are just variations of their name. While this isn't always the case (as seen by Charizard, who needs no name introduction), it usually is because from a marketing standpoint they want people to be as familiar with the franchise as possible. I don't agree with it from a lore standpoint but thats just good marketing...
You might notice in the 'Pokemon Origins' animation, which is a more serious take on the story, the Pokemon don't say their names at all.
Talking more specifically about Pokemon represented in Smash Brothers, they obviously use more than 4 moves just when looking at their normals. Even Greninja's Substitute special move is actually a combination of two moves because of the follow-up kick attack.
I just assign names to all the attacks the Pokemon uses because its fun and shows part of where I'm taking inspiration from when giving them the attack. I try including as many moves from the primary moveset and try hard to stay away from anything that can't be attained in any way in-game sense it gives parameters without being overly limiting.
+ Retrospect
I understand more clearly now you were probably just thinking what moves to best represent Cinderace or one of its forms in your Galar Trainer moveset but hopefully this was all helpful and my light-hearted rant on 4 moves wasn't too overbearing despite perhaps misunderstanding your original intent.
I'm curious, what's your general idea for the character? Do they use the first, middle, and last form just like the Kanto trainer?