I'm on Debian Wheezy stable with Gnome 3.4 and have several default extensions, which on the 'Installed Extensions' page at https://extensions.gnome.org/local/ (using Iceweasel's Gnome Shell Integration plugin), are not able to be removed like the ones I have added from extensions.gnome.org myself.
These include (for me anyway - and this may not be the complete list as I probably have one or two of them enabled and thus no quick way to know all of them):
Alternative Status Menu
Applications Menu
Auto Move Windows
Dock
Gajim IM integration
Places Status Indicator
SystemMonitor
User Themes
windowNavigator
Workspace Indicator
Inspecting the HTML of the Installed Extensions page, I see no way to hack it via Firefox's developer tool (at least with my limited HTML knowledge and only from a quick look) to somehow make those extensions uninstallable via that page.
Is it as simple as deleting (e.g. for the first one) the folder /usr/share/gnome-shell/extensions/alternative-status-menu@gnome-shell-extensions.gcampax.github.com, then restarting Gnome Shell and it'll be removed from the list?
I want to do it correctly, without introducing problems later.
gnome-shell-extensions. If you don't use gnome-classic I think you can remove the package. Otherwise you'll have to remove the extension directories, as you said, but they'll get reinstalled each time you update your system. The web-based installed cannot remove these extensions as it's limited to the ones installed in your~home.