So, to finish this thread off, and for those other Newbies, here's the correct answer:

In the Preferences pane (pain) in the first window that opens, named "Interface," SCROLL DOWN to the bottom of the window. (5 days of hell and all I had to do was scroll down!) Under "Devices" there is "Mouse Tracking." Deselect the radio button named "Scroll Wheel" and Select "One Button."

Now when you hold down <ctrl> ("Control") and hold down the one pathetic button on the mouse, you will get that heavenly pop-up option to select Vertex, UV, Faces, etc.

Thanks SO much to everyone, especially to gster123 and dannyngan.

BTW, I promise to be more careful where I start my threads!