Mid-2010 MacPro not upgrading to HighSierra.

I have exactly the same model, and yes, High Sierra will install on your Mac.

First check macjack's suggestions to see if the High Sierra installer is still in the Applications folder. Download it again if necessary.

A major point with this model, High Sierra will not install right away. The first thing the installer should do is inform you the Mac needs a firmware update. This must be done so it will be able to see, read/write, and boot to an APFS formatted drive. Once the firmware update is finished, then High Sierra should install.

Yes, I ran into that loop with the Mojave beta, though the reason may have been different from yours.

My issue: I had created a new, empty partition on my SSD to install Mojave to. Being in beta, things don't always do what you would expect. The installer told me right away that Mojave had to be installed over an existing OS. So, no "clean install" available yet. Okay, so I booted to another drive and cloned High Sierra to the new partition so I could upgrade one and still be able to boot into High Sierra.

As soon as the clone was finished, I started the Mojave install from the drive I was on, and chose to install it on the cloned partition. The screen for the firmware update appeared (there's yet another one for Mojave). It went through the motions of shutting down, and I held the power button, which flashed and beeped. Like you, it sat for a bit, then booted back to the drive I was last on without appearing to do anything.

Turned it, it simply didn't like the way I was trying to apply the upgrade. For whatever reason, the firmware/OS install sequence only works properly if you are installing Mojave to the same partition you booted from when you start it.

Your issue: It appears you must be logged into the Admin account instead of a standard user account to install High Sierra. Likely do to the higher level of necessary permissions to install the firmware update. Yes, you can enter the same admin name and password from a standard account if you know it, but it's still not the same as actually being logged into the admin account.

Restart and login to the admin account. Go into the System Preferences and change the automatic login to the admin account to make sure the installer doesn't get confused by starting from admin, but booting back to a standard account. Then, try installing High Sierra again.

Then I’m afraid I’m out of ideas. Makes no sense the firmware won’t install.

Okay, I lied. I have one more idea. Create a new partition. Download Sierra through the App Store. Install Sierra as a fresh install onto the new partition. When the install is done, it will become the default startup drive.

Now try installing High Sierra. It’s just a thought to see if something on your current startup drive is somehow interfering. From a fresh install of the OS with nothing else, it should work.

If it is the white plastic version of the Radeon 7950 it is the same as mine and I was able to install both the High Sierra firmware upgrade and the Mojave beta firmware upgrade.

The process on older Macs like the Mac Pro is different to newer Macs like the MacBook Pro. You run the High Sierra installer, it says it is going to do the firmware install, however on the Mac Pro it installs it (or should) and then requires you to shutdown the Mac completely. You then turn it on and keep the power button held down until the light flashes rapidly. You then let go and it should do the firmware update.

On a newer Mac you simply let it install.

You could try the following although there is no reason why it should work instead since it will in theory be using the same firmware installer.

Previously some people wrote a script to extract the firmware installer from the High Sierra installer so it could be run separately, this was considered useful for building Macs in an enterprise environment using imaging. I recently made a new version of this script because 10.13.3 and later broke (for me) the original script.

You will need to have the High Sierra installer app in your Applications folder as normal, you then run this script and it will extract the firmware installer and make a standalone installer from it.

As per the ReadMe my script needs the https://github.com/munki/munki-pkg tool, this is a python script. You simply download the munkipkg file from that github page, you need to save it with the filename munkipkg and place it in /usr/local/bin you need to do chmod +x munkipkg to add execute permission to it. You then also download my shell script from my github page at https://github.com/jelockwood/extract-firmware and save it anywhere you want and also give it execute permission as well. You then run my script which will use the munkipkg tool. The resulting installer will be produced in /tmp/FirmwareUpdateStandalone

You can get to the above folder by using the 'Go to Folder' command in the Go menu in the Finder.

If you can run Sierra, then you should be able to run High Sierra.

Have you tried installing in Safe Mode:

Use safe mode to isolate issues with your Mac - Apple Support

It's also possible that you downloaded it but that it did not install. Look in your Applications folder for "Install Mac OS High Sierra" and if it's there double-click to install.

Shouldn't. I just recently put an EVGA (Nvidia) GTX680 Mac Edition card in my 2010 Mac Pro so I could run the Mojave beta. I'm using an OWC Mercury Electra 3G SSD I put in about five years ago, also purchased from Other World Computing.

Basically, I can't see why either would be a problem. By default, once things are figured out here, you should be aware that your SSD will automatically be converted to APFS (assuming it's the current startup drive you are upgrading to High Sierra). Just saying that to reiterate that you should backup that SSD before installing High Sierra. APFS has caused no problems for me at all. It's simply prudent to have a restorable backup before installing something as major as an OS upgrade.

I attempted what you suggested with a new partition. I got stuck in the loop again. Requested me to log in as an administrator even though I already was to install the helper. Same thing over and over.

When you can't create a partition from an admin account, something is definitely wrong.

Do I need to do anything silly like unplug peripherals from USB ports or FireWire ports or the audio jack? I installed a PCI card to give me USB3 capabilities, would that cause a conflict?

With third party peripherals, there's always a possibility. It's a worth a try to minimize the attached hardware. Remove the expansion card. Unplug any other devices so all that's left is the monitor, keyboard and mouse. See if that allows you to proceed.

If not, I would make a full, restorable backup, then reinstall the current OS.