The screen on my 2012 iMac stopped working, and does not respond to the usual quick fixes. How can I display the iMac and its menus on another computer to explore other diagnostics? On a Mac Pro?
Try an external display. If that doesn't work at all then your display controller may have died. If you do get a display then your internal display panel may have died.
I had a display failure last year on one of our 2012 iMacs at work. An external display worked fine. The key to resolving our issue was it came up as the only connected display - the iMac had lost communication with the internal panel. The display controller clearly was working as it was happily driving our external monitor. Replacing the panel was the solution in our case.
If you still wish to perform diagnostics (and working on the assumption that your iMac doesn't have Screen Sharing, Remote Login or Remote Management enabled in your Sharing preferences) your best bet may be to use another Mac to create a temporary OS X installation on an external drive. On this drive ensure Screen Sharing is enabled.
Then, start your iMac with the external drive connected and hold down the Option key to bring up the boot selector - which you obviously won't be able to see on the internal display. Press the right arrow key once and hit enter, and all being well your iMac should startup from the temp OS X installation.
Make sure your second Mac is connected to the same network as the sick iMac. The iMac should appear in the Shared list in Finder and you will be able to initiate a remote session via the Screen Sharing option.