The disappearing addon icons was how I found this issue in the first place.
I have a folder at Kodi root that I keep some icons and things in for my skin setup, and since this can be referred to as a special path folder too you can set it in the skin settings in the same way and it will work across all your devices. My folder is called "# art icons" but you can name it how you want.
So you need to go through and edit the skin settings for these addon icons too after you have made them, and point it to this new folder that you will make.
An example path would be special://home/# art icons\MyAddon TV Shows.png
(you need to edit your previous post to remove reference to this banned addon)
Change anything after special://home/ to match your particular folder and icon names.
Regarding your universal libraries, they can point to a special path too, as long as it's inside the Kodi root folder somewhere.
Or, if none of your devices will be taken outside the home, point to a network folder if your addon supports that and have one centralised folder for all your devices.
Normally an example path for what you are referring to would be: special://userdata/addon_data/plugin.video.myaddon/TV Shows
But per my above example you could also make your own folder and point it there, eg: special://home/TV Shows
I guess this is what you have done already, but you are pointing to it with a local path, instead of a special path. So just correct that in your addon settings.
Hope this helps