There are a few ways to install and manage your extensions.
-
The default Extension app (aka gnome-extensions-app) is part of gnome-shell and thus installed when you install GNOME. You couldn’t use this app to install extensions but to manage them (their settings, removal, automatic updates).
-
The “traditional” way of installing extensions via the website: extensions.gnome.org
In order to install extensions from this site you would need to install:
and Gnome Shell Integration addon if you use Firefox (or similar for Chromium-based)
- The “modern” way of installing extensions with Extension Manager (flatpak) or the AUR package
You could both install, remove and manage your extensions with this app
- The manual way:
https://www.debugpoint.com/manual-installation-gnome-extension/
Any extensions installed via Web page, using Extension Manager or manually will get installed locally under your home directory: ~/.local/share/gnome-shell/extensions
Any extensions installed from the repositorie using pacman/yay will be installed system-wide in:
/usr/share/gnome-shell/extensions
These have gnome-shell as dependencies, so removing GNOME thus removing gnome-shell should consequently remove these as well.
Most probably I have missed some detail but I think this covers almost the “long and short” of it.
This is unlikely to happen 
