In Gen 3 PID, IVs, nature and gender are related to eachother. If you change any of those values you have to calculate a valid PID for them, PKHeX won't do that for you.
Your save is gone unfortunately. That file doesn't look like a save file at all, there's only junk inside. Maybe a hardware failure of the Analogue Pocket?
This has been discussed a few years ago
The problem is that the original OT is too long. It's 7 characters but the Korean language only allows 6. When you receive a Pokemon with an invalid OT (which is either too long or including a censored word) the games will change it to a default one.
There is nothing you can do about this. GF doesn't know their own games so they messed this one up unfortunately.
PKHeX only fixes some minor issues when you view the Pokemon. But you can not use the Batch Editor for that. The Valid ad Legal properties are readonly, you can not modify them.
If you want to auto-legalize multiple Pokemon you should use ALM https://github.com/architdate/PKHeX-Plugins/releases/latest
What exactly does not work?
You can get the latest automatic build artifact from the azure pipelines. Works fine for me on the latest PKHeX (230925). https://dev.azure.com/architdate/architdate/_build/results?buildId=937&view=artifacts&pathAsName=false&type=publishedArtifacts
Well, you could turn on HaX mode in PKHeX to give a Pokemon the GS Ball, but that won't work.
Having the GS Ball alone is not enough. There are also 2 event flags that have to be set. For that you have to edit your save.
That's because the Pidove was never available like this. These wondercards were never meant to be accessible by anyone, they were used for debugging by GF.
You shouldn't expect anything from the Unreleased/Beta PKM category to be legal. Some might be, but only because they match another Pokemon that was actually available.
You shouldn't open the .bat in PKHeX. If you double click the .bat PKHeX should open in HaX mode.
This guide is a bit out of date. Nowadays you could as well just set ForceHaXOnLauch in the startup settings.
Impossible.
Well theoretically it would be possible. The game unfortunately has many flags/variables that depend on the gender, so changing the gender without changing all those flags would break your game. There are hundreds of them and no one has figured them out. That's why changing fhe gender is disabled and not recommended.
Directy dragging the save on the exe won't work. PKHeX doesn't tell the plugins that a save was loaded.
So you need to open PKHeX first and then drag your save into it.
Edit: fixed in the latest PKHeX dev build.