Yes, but I solved the problem. FINALLY!
On some Macs, 10.8.5 installed a visible file at the top level named: mach_kernel, which should have been installed as an invisible file. On an unrelated Mac problem, I was asking an "expert" from AppleCare if I should delete this mysterious file. As I had never seen such a file in all my computing years, I was skeptical of deleting this file without consulting Apple. The "expert" told me to delete it as it was not a serious file, which I did.
Related to all the problems of installing Flash Player updates as described above, further Google research indicated that I should try to find the original "mach_kernel" file and restore it. Fortunately, I could locate this file on a backup disk, which I reinstalled on my Mac HD. The file copied quickly and instantly transformed itself into an invisible file, which it should have done in the first place.
Then I tried to install the latest 12 version of Flash Player Installer and, Voila!, Flash Player now works on all browers.
Thank you Mike M for your initial offer to help.
Google is your friend.
That's exactly what I was prepared to suggest to you (replace the mach_kernel).
Glad yoiu got it worked out.