Chosen Solution
I replaced the GPU (many thanks to the well done ifixit.com tutorials) and the Logic Board Battery after a very long inactivity period. I made a a clean OS X installation (El Capitan) and now both my iMac and the new GPU work well after the user login, anyway I cannot see any video signal before the user login screen. All the 4 diagnostic LED are working I checked, but I can’t get the HD selection (option command) at the machine startup - or maybe I can but I cannot see it. I’m baffled, what is that strange behavior due to?
This is the expected behavior when using a non-Apple GPU board. The board does not have the custom Apple firmware.
