Hello, I just recently downloaded Avast! aswell, so I know the settings and all that. I'll help you with your first question and then, if I fix that, I'll proceed onto the next.
1.) Access the program from the start menu (Unless it's on your desktop, then just double click it).... "Start", "All Programs", "avast! Antivirus" and then again "avast! Antivirus". It should be starting a memory scan, wait until this is complete. After this is done the actual Antivirus scan page will show up. Hover over the top left symbol that looks like an "Eject" symbol. It should say "Menu - Show the program menu". If so, click it. A list of options should now appear, go to the one that says "Settings" and click it. While in Settings, click "Appearance". For your settings, the box that says "Show avast! tray icon" should be unchecked, you want it to appear in the tray so check that box, if the other one is not checked you can also do that. This should hopefully solve that problem. Tell me if it does so I can help you with your next question.