Im sending this email again to gnome-shell-list because im new to mailing lists and I think I might be replying only to Leslie.
Leslie, apparently extensions save their settings in gsettings, which can be accessed by dconf.
To save a copy of the settings you can do:
dconf dump /org/gnome/shell/extensions/ > settings.txt
To load the settings:
dconf load /org/gnome/shell/extensions/ < settings.txt
So maybe a bash script can be made to automate a tar file with the extensions and settings.txt. And another script to install the extension and the settings.