I will deal with what is probably your main concern first. The avast icon is just the interface to on-access scanner and menu selections for settings, on-demand scans, etc. a pain in the rear but not directly related to protection.
Check Task Manager and look for the avast processes, they begin with ash or asw, what do you have running (see image) ?
Do you have teatimer running in Spybot S&D as this could be removing the resident protection or if the option exists to exclude ashDisp.exe from detection. Is the AdAware the free or paid for option, if paid then AdWatch could be the issue. If none of those then any of the most common security programs listed below ?
As a temporary measure until this is resolved you can create a desktop shortcut for this file C:\Program Files\Alwil Software\Avast4\ashDisp.exe (the avast icon and interface to the providers). Right click on the file and select Send To, Desktop (create shortcut). You will need to run this after each boot until the problem is resolved.
What other security based software do you have that might block new startup entries, e.g. SpySweeper, Spyware Doctor (StartUpGuard), PrevX, WinPatrol, ProcessGuard, etc. ?