I've changed it so that the submitter's email address is displayed as an additional column in the Unanswered questions area. In addition, I've linked the email address with a simple "mailto" tag so that I can simply click it to reply.
The email automatically has the Subject line filled in, and the person's FAQ question is quoted in the body of the message. All I need to do is add my reply and send it off.
These changes only required slight modifications to the adminfaq.php and global.php FAQ language file. If anyone wants the changes for their server, you can download the zip file here