Yesterday, based on your suggestion, I checked the "Disable internal pointing device when external USB pointing device is attached" (under Mouse Properties / HP Control Zone Settings tab). While hitting "Apply" causes an eternal hourglass and entering OK greys out the window, the laptop's mouse nevertheless is disabled. And strangely, today, the touchpad came back to life suddenly and I had to repeat the selection.
But... when the touchpad/mouse is disabled, I have had absolutely no problems with the cursor moving suddenly and hapazardly etc. Your idea was right on the money. Thank you very much for solving my problem - I really appreciate it.