Cause people routinely manage to put somewhere other than .minecraft/ffmpeg/bin/ffmpeg.exe.

It'll search the folder which MC uses to store all its stuff (MinecraftClient.runDirectory). So, yes to the first question, no to the second one.

Does this work for other launchers, e.g. when my instance is in c:\Users\lyn\AppData\Roaming\gdlauncher_next\instances\FTB Interactions 2.0.9\?
If my instance is in a different launcher's instance folder, will it still also look for ffmpeg.exe anywhere in c:\Users\lyn\AppData\Roaming\.minecraft\?