Drew,
Here is my successful experience with MS Tech Support. It is a little long, but I figure you are probably interested.
First off, I ran DISM.exe ... RestoreHealth. It came back with "Error 0x800f0906, Source files could not be downloaded. Use the Source option to specify the location of the files that are required to restore the feature."
So I called MS Tech Support and, surprisingly, despite the heavy load warning, I got through in a couple of minutes. I explained the story-so-far and was forwarded to a technician (in Jaipur) who seemed familiar with the issue and assured me he would fix it.
I tried to follow what he was doing, but much of it went by too quickly. He did a couple of things which caused the PC to reboot a couple of times. He also downloaded an MS Support Diagnostic Tool. Then he ran Action Center as I had before. AC fixed two issues.
He then ran DISM.exe ... ScanHealth, which came back with an issue and message "component store is repairable". Then he ran DISM.exe ... RestoreHealth. So things looked good. After this ran, there appeared to be an issue which was not fixed. He did not pause here, so I was unable to see what it was and when I asked him about this, he said not to be concerned, it was ok.
He then downloaded and installed 13 updates, rebooted and it came up with the message I was getting before! "PC ran into problems". The PC tried to auto-repair itself, with two reboots, but as before, it came back with "could not repair' and gave me the usual options (proceed to Win8, shut down, or Advanced Troubleshooting). We selected the third option and (as before) were given three options: Refresh your computer, shut down or other options. (I had selected the third option twice previously and ended up doing a System Restore). In this instance, the technician told me to Refresh the computer. We discussed this a little, but he was confident to go ahead.
After the Refresh and reboot, he said everything was solved. I told him I would not be assured until I had gone through a couple of updates. Well I went through the first round yesterday and it went smoothly, so, so far, so good. With all of the reboots, scans, refresh, it took over 4 hours!
I am not sure if any of this is useful or interesting to you, but I thought I owed you the end result for your much-appreciated assistance. So thanks once again.
best regards,
Moofus21