If anyone is remotely interested, I finally solved the problem. My guess is I could have reinstalled the main OS. Instead, I upgraded it to Windows 10 and then installed Player 7.1.2. On the plus side, no more problems with 2 mouse pointers with one randomly showing up where and when it wanted.
On the negative, I now have to always press Ctrl-Alt to get to the VMWare bar at the top of the window. But this is a small price to pay for now being productive again....
FWIW, my guess is that upgrading to 7.1.2 is really what made the difference.