Have you had any other AV installed on the system?
This is often the most common reason for it not being installed to avoid conflict with another AV.
The email scanning icon is only there for the duration of the email scan, have you ever seen the avast a icon in the system tray?
Try a repair of avast. Add Remove programs, avast! Anti-Virus, Change/Remove button and scroll down to Repair, click next and follow. You need to be on-line to do this. This is not usually enough to restore the icon.
If that doesn't work try, uninstall, reboot, install, reboot.