no worries no worries. Just treat it as an explainer for future readers or something
yeah save in-game, soft reset the game, and if you can "continue" from the title screen without using any state saves, then it's probably saved in-game. There, close the emulator, then drag the sav into PKHeX.
If it still doesn't work, then maybe it's saving into a different folder or something. Check your settings, search your computer for the save.
Well, the only way to find out for sure, if play the game on a flashcart on your GBA (if you have one), and see if it's still as laggy.
no idea what is BizHawk. From a quick google search, it's an emulator for TAS?
In any case, there has to be a folder it saves the save into. And most saves aren't really that different, as long as it saves into the correct size. Since I don't use it, I can't give proper advise on that, beyond check your settings, look for save destination folder, and check if the save is in there (when the emulator isn't open).