There are two types of Add-ons. There are the Extensions that you find and install. And there are also Plugins. These are from other programs that are on your system when Firefox starts. Like video players, anti-virus, and many more. You can disable normal Add-ons by going to the Add-ons Manager,

Data from extensions are stored in the storage folder (indexedDB). However this extension data is linked via a UUID that is stored in a pref in extensions.webextensions.uuids in the profile folder, so it is not easy to transfer extension data. Some extensions offer to backup their data, so first check that if you still have access to the old setup.