I have a couple of different iMacs from that time period. I often get intermittent results when using various boot key shortcuts. Usually, that is helped by doing an NVRAM reset during first boot.
Of course, the Option boot picker window is always the Option key (no other key pressed) (I guess that is what you mean by "Disk Choices menu" You have to use a drive that you know has a system installed that will actually boot your iMac. Some older Macs will bring up that menu, and will NOT show any bootable device that is not compatible with your older iMac. Your 2010 iMac should show any system from Snow Leopard (OS X 10.6.3) to High Sierra (macOS 10.13.6)
That bootable system needs to be correctly installed (or external drive actually plugged in to your USB port.)
I still think you will have an easier time of this with a bootable USB Mac system installer.
Your boot with Option+D should be booting to hardware diagnostics, and should not work unless you have an internet connection. And, I have found that it is very common for that to fail with an error (I think that Apple does not support Diagnostics for Macs that are considered obsolete, typically those more than 6 or 7 years old -- so yours is considerably older than that.
If you try to boot to internet recovery, try the default choice for that, which is Option+Command+R
You should get the spinning globe. Best choice will be wired ethernet (test without using wifi)
That boot to internet recovery can be very slow. 5 minutes is pretty fast. I have seen a few that eventually boot after nearly 20 minutes. Again, you should see a spinning globe. The default Apple icon might appear for a few seconds, but internet recovery boot should always show the spinning globe at some point - might take a minute to see that, but you should see it. Don't use any other keys, only Option+Command+R
If you only see a blank screen, and it has been longer than 5 minutes, press and hold the power button to shut your iMac OFF, then try again. Press and release the power button, then immediately press and hold Opt+Cmd+R. Do not release the keys until you see the spinning globe. Good luck!
If the boot fails, you may get an Apple support.com error of some kind, which USUALLY means that there is something wrong with the internet connection. I will often try a simple restart of my router, if I get an Apple Support error, then try the same Internet Recovery again.
Finally, if you DO get to Internet Recovery menu screen, go to Disk Utility, and try to erase your drive that you have installed in your iMac. Erase the drive, choosing to format as "Mac OS Extended (Journaled)". I suggest that you don't mess with APFS format. With the problems that you have, you may need to install an older version of the system first, and you may find that the Disk Utility in an older system will NOT see a drive that is formatted APFS. So, go with the older format for now. When you get ready to install High Sierra, the installer will change that older format to APFS automatically. I have done dozens of reformats, and High Sierra (or newer) never have a problem with the format change. That might even help you out as you try to get your old iMac fixed up, and upgraded.
forums.macrumors.com
