Thanks for the reply!
I did another reboot, after disabling a couple of start up programs that were displaying their icons in the lower right of my taskbar.
I now have only the system icons (clock, volume, network, etc.) appearing besides the orange/blue ball Avast icon.
I disable the programs from starting up with Windows 7 Ultimate (x86) by either right-clicking on its icon in the taskbar and disabling it from starting up again, or running msconfig and disabling the program from starting up from there.
That appears to have done the trick. After about 20 seconds after reboot my orange/blue ball appeared, and I now have access to the Avast anti-virus program, again. And, like the label on the Avast ball icon says: "Everything is good."
.....
P.S. BTW: This is a slow Athlon XP 2000+, 2 GBs of memory, and an old Direct 9C AMD Radeon 9600 Pro video card running at a 4x AGP interface. Barely meets specs to run Windows 7 Ultimate.