After doing even more research on all of this, I've come to find that not much of anything really works very well. Even after trying the different combinations of files I still can't always get a computer to run its updates. For instance I have a Dell Latitude E6510 that is updating (it appears) without any problems by using this combo (in this order):
KB3020369 - April 2015 Service Stack
KB2670838 - Platform Update
KB3125574 - April 2016 Service Stack - this is a good-sized file and will take some time to install.
Install each in order as above (with the unit connected to the Internet), restart the computer after each install before proceeding to the next update / file.
So the Dell is working great - taking all three of these updates without any problems.
But I have an HP Workstation that I'm attempting to do this very same process with. (I might mention that both of these computers have had a fresh load of Windows 7 64-bit today.) The HP is NOT taking the updates - for some strange reason. I have shutdown the HP and have restarted it a couple of times but it has made no difference.
So what gives I wonder? Other than I truly believe that Mickey$oft is really attempting to discourage anyone from using Windows 7. Sorry to tell you M$, but most people around here don't wish to make the upgrade to W10.