I have no idea how or why these registry permissions are a problem, but I do
know that the following actions resolved the problems and allowed Flash Player
to properly install for me.

I opened regedit, and searched (using "Find") for entries that had keys,
values, or data containing the word "flash". This returns a lot of entries,
including a lot for "aolflash". I ignored all the AOL ones, but all the rest I
right clicked on the tree item and selected "Permissions", and checked the
permissions for the Administrators Group. A number of the registry entries did
not allow the Administrators "Ful Control" and "Read" permissions. Where those
permissions were not allowed, I checked the Allow box so that they would be
allowed and clicked the Apply button.

After making sure that the Administrators had "Full Control" and "Read"
permissions for all the flash entries, I then exited Regedit and ran the flash
uninstall program one more time, then went to a flash page, where I was
prompted to install. I chose Install, it did, and then the flash player
played. SUCCESS!!!

Now, the warnings!!! Messing with your registry can really screw up your
computer to the point that it cannot be run again without reinstalling
everything from scratch. If you're not comfortable backing up the registry and
editting it, DON'T MESS WITH IT! Get someone who knows what their doing to
help you. I don't know which entry or entries were the important ones, so
you're on your own, but maybe this info can help Adobe to figure out a safe
fix. Anyway, good luck. Thanks to everyone who offered advice along the way.