First, how did you remove Norton? (You should uninstall via Add/Remove programs, then use their uninstall tool and instructions: http://us.norton.com/support/kb/web_view.jsp?wv_type=public_web&docurl=20080710133834EN&ln=en_US)
What version of Outlook? Do you see all of your data files in Control Panel > Mail > Data files?
Avast only loads a plugin in Outlook to scan messages as they come in or go out. It shouldn’t hide folders or delete them in any way.
You could try loading outlook in safe mode to prevent the Avast plugin from running and see if your folders show up (run Outlook.exe /safemode) from the run box.
Or, you could try repairing your PST file if you aren’t on Exchange http://support.microsoft.com/kb/287497