Hard Drive partition mounting problem

The partition you are trying to examine is an Apple File System (APFS) partition. This was first used on the boot SSD under 10.13 High Sierra, then propagated to other dives later.

If the version of Disk Utility is from an OLDER Recovery partition, it will not be bale to understand APFS partitions. You will need a more recent disk Utility to do that checking.

what Macs and what MacOS do you have on hand?

One possibility is to make a Bootable USB Thumb drive installer for 10.14 Mojave. if you boot from that, its Disk Utility would be able to check the APFS drive. There is no need to use its installer.

Your Mac Pro 2010 may need to have been running 10.13.6 at some point to get all the required firmware updates to be happy going forward. (Firmware updates are stored in a Private store on the processor card, NOT on the drive).

Disk Utility in 10.13.6 should be able to repair an APFS disk, but it possible some details were overlooked. Mojave Disk Utility may be "smarter", which is why I suggested you could use the one from a Mojave Bootable USB-stick installer drive (without actually installing Mojave). Post back if you want more detailed instructions on making that. Requires a 16GB USB thumb drive that can be completely erased.

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Another possibility is that the drive has very extensive damage, and you may need Data Rescue or a similar program to help you search for files. These work by making excursions onto an unmountable drive to look for patterns that resemble files. Requires a different drive to store recovered files.

Another Utility that maybe helpful is Disk Warrior. Both of these cost money, and provide no guarantees of good results.

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By far the best way to protect your data is using the built-in Time Machine to make quick incremental backups on an ongoing basis.

I would boot into Yosemite and use its Disk Utility. I've used it in the past to countermeasure difficult drive problems. No guarantee it will work but what else to do.

Disk Utility in Yosemite is FAR too old to repair APFS partitions.

Sierra MIGHT be able to, or it may struggle. Definitely nothing older.