I'm not sure which web interface you are using (AsteriskNow, Trixbox, FreePBX, or some other web interface). I know that AsteriskNow will modify the extensions.conf file when you add extensions, or incoming/outgoing features to the dialplan. BEWARE, if you edit any of the .conf files yourself that AsteriskNow also writes to, some of the web pages will not display properly if certain elements are changed or eliminated by your editing. Dont ask how I know this!!! :)
To the best of my knowledge AGI programs are not changed through the web interface. They are called from the dialplan with the AGI() application, then dialplan processing is then handled by the AGI program, when the AGI program is finished processing, it returns control of the dialplan to Asterisk. Of course AGI scripts are used when Asterisk's capabilities are too limited to do the kind of processing required within the dialplan itself.