The “Copy Render Settings” Add-on is only to copy settings between scenes and you need to checkmark, what settings you want to copy.

“Save Render Presets” makes your render presets available in every Blender file and it always saves all settings to avoid unwanted changes.

Yes, you can. It's just important that the path you entered in the Save Render Presets add-on settings is the same in all blender versions. Also note that some settings are not available in all blender versions. For example, blender 3.0 has different Cycles samples settings than blender 2.93. Logically, blender 3.0 cannot save the samples settings for 2.93 because they are not available.

You can find the old presets here:

Linux~/.config/blender/2.81/scripts/presets/

Mac/Users/{user}/Library/Application/Support/Blender/2.81/scripts/presets/

Windows C:\Users\\AppData\Roaming\Blender Foundation\Blender\2.93\scripts\presets

Because the user can now enter a path himself, this problem is now a thing of the past.

And yes: the old presets are compatible. Note that some settings are not available in all blender versions. For example, blender 3.0 has different Cycles samples settings than blender 2.93. Logically, blender 3.0 cannot save the samples settings for 2.93 because they are not available.