I finally solved this problem!
Through trial and error, I have discovered that the most recent Adobe Flash Player version that works on my desktop is v. 18.104.22.168 (released Mar. 5, 2012). Anything more recent than that version does not work!
Could it be that newer flash player versions require more recent hardware to run? (My desktop is 12 years old!) I do not know. Perhaps somebody on this forum knows more details.
I am aware that using an old version of flash player is dangerous. But I am not going to avoid certain sites that require flash player. I will be as careful as possible! :-)
PS 1: I installed libflashplayer.so in /usr/lib/mozilla/plugins
PS 2: Older versions of Adobe Flash Player can be downloaded from http://helpx.adobe.com/flash-player/...-versions.html