@doodledoctor,
Please try the following (you may find little changes according to the specific version of Windows):
1_ Task Manager -> Services -> Avast Antivirus -> Click on the lower button "Services". Choose from the new list of services the one for Avast Antivirus. Right click -> properties -> Startup type SHOULD be "automatic". Please be sure it is (and if not, change and save it, must reboot and review this same setting).
2_ Now Task BAR Properties -> Notification Area -> CHECK "Hide inactive icons" -> Customize -> Select "Avast Antivirus" -> Behavior (column) -> Select from the options: "Show" -> save all options including the properties dialog of the task bar is closed with ok.
3_ Close all your opened programs. Change your resolution (temporarily) to 800x600, still with color quality of 32 bits, and (if your screen display lets you, set the refresh frequency to some value between 60 to 100 Hz (meaning, standard "low" resolutions. Your desktop appearance will change (a lot). Don't panic! And don't start moving any icons. All will be as it was when you change again to your original resolution after the test.
Take a look at the tray area. Do you see your Avast icon under these conditions? If still not, try changing also the color quality from 32 to 24 or 16. Take a look at the tray area. Do you see your Avast icon under these conditions?
4_ Reset all your display settings (resolution, color bits, refresh rates) to the previous values you had. Your desktop should return to show the icons as they were before (if you haven't moved them).
(Note: the resolution of 800x600 is very low for your screen, and it uses a very different relation than yours. It may be difficult to really see something, and it may require from you to temporarily change the settings of your screen from its menu - not only from the graphics settings of your OS. As said, don't panic, be patient, and all will return to be displayed normally once you return to your normal resolution.)
Please report back your results.