Okay. Uninstalled 4.6, rebooted (sigh), installed 4.7 from freshly downloaded installer, rebooted (sigh...). This time Avast started, gave me the 'Welcome to Avast' screen, and the 2 icons in the systray. The main icon had the little red o on it, though. I clicked on it to get to the registration pop-up, and Avast crashed, the two icons disappearing a moment later.
I went into the Avast4 folder to see how the permissions were set. As they were the last time, no user names and no checks in any of the boxes. So, I manually set all of the file permissions (selecting them in groups and by folders, of course) to Everyone- Full Control. Odd thing is that there were a couple of folders that did already have permissions set (the same settings), but most did not. Now they are all showing the same settings (I think) as they did under 4.6, with the only difference that they are solid and not grey checkmarks.
I double clicked the desktop icon, entered the reg code, and the program ran fine.
I rebooted (... sigh...), and now everything seems to be functioning again.
I only worry that I will have to go through this process with the next update as well.