Hi 
I know the date/time doesn't seem like it should matter, but it does. There are several reports of changing the date/time to "auto synch" helping with Origin issues. Right-click the date/time in bottom right of screen > choose "adjust date/time" > "region" (left side of screen)
The easiest way to check if it's the issue is just change the settings under Time & Time Zone to "Automatically"
We can hold off on the DNS flush for right now, I understand your worries. It looks scarier than it is. But, let's try some more connection troubleshooting before moving to that.
Obviously, your computer is online or you wouldn't be able to post here, so make sure you haven't accidentally put Origin offline - in Origin, go to the Origin menu > Go Online (if it says "go offline", then it's already online) . Actually, with all of the different work you've done, I would just Exit Origin completely - Origin menu > Exit - then reboot your computer. After your computer loads Windows, log in to Origin. As long as it doesn't tell you you're offline - right-click SimCity. If there is a "Repair" option, choose to repair and see if we can get it installed.
If there is no Repair option, please send a screenshot of what the menu looks like.
Thanks for hanging in there with the troubleshooting - Origin can be stubborn sometimes.