Its just the step by step process is different now, the reply 3 from Pondus shows how it is now, at the bottom of the popup it says customise which takes you to a screen with the option on what to do with all the icons.
From what the op has said ( It's not hidden - I've set it to "show icon and notification" on the tray settings. And clicking on the little arrow does not show it.) he's done the correct process so i believe dvirkk needs to do a clean install as Gargamel360 has explained in reply 1.
Thanks for the remainder about the "properties" of tray area in Seven. Independently from the exact Windows version and the exact steps, my point is to try to explicetly set the Avast program as permanently in "show" status, instead of leave it as Windows "decides" to show it or hide it.
The problem might be related to Windows tray icons cache, so (re)instaling would not solve the problem in such a case.
In addition to set the Avast program (whichever icon is shown in the tray properties, doesn't matter), the OP can try cleaning the Windows tray icon cache:
Regedit
Vista/seven:
HKEY_CURRENT_USER\Software\Classes\Local Settings\Software\Microsoft\Windows\CurrentVersion\TrayNotify
XP:
HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Explorer\TrayNotify
Delete the IconStreams and PastIconsStream values
log off immediately.