This may be stating the obvious, but the email scanner will only scan emails for the default email program on your computer. So if you have more than one mail program on your computer, check that the one you use is set as the default. Obviously it won't check webmail, but webmail sites generally have a spam and virus scanner built in.
Open the AVG user interface, double click email scanner and see if scan incoming, scan outgoing and display notification window are ticked.
As for the other problem, if rebooting does not solve the problem you might want to uninstall AVG and reinstall it.