High Sierra-Lost Permissions-Boot HD

Permissions for your SSD are proper and you should not attempt to change them. See the following:

Possible Fixes for El Capitan or Later

You should try each, one at a time, then test to see if the problem is fixed before going on to the next.

Be sure to backup your files before proceeding if possible.

1. Shutdown the computer, wait 30 seconds, restart the computer.
2. Resetting your Mac’s PRAM and NVRAM
3. Reset the System Management Controller (SMC)
4. Start the computer in Safe Mode, then restart normally. This is slower than a standard startup.
5. Repair the disk by booting from the Recovery HD. Immediately after the chime hold down the Command and R keys until the Utility Menu appears. Choose Disk Utility and click on the Continue button. Select the indented (usually, Macintosh HD) volume entry from the side list. Click on the First Aid button in the toolbar. Wait for the Done button to appear. Quit Disk Utility and return to the Utility Menu. Restart the computer from the Apple Menu.
6. Create a New User Account Open Users & Groups preferences. Click on the lock icon and enter your Admin password when prompted. On the left under Current User click on the Add [+] button under Login Options. Setup a new Admin user account. Upon completion log out of your current account then log into the new account. If your problems cease, then consider switching to the new account and transferring your files to it - Transferring files from one User Account to another.
7. Download and install theOS X El Capitan 10.11.6 ComboUpdate or 10.12.6 Combo Update or macOS High Sierra 10.13.2 Combo Update as needed.
8. Reinstall OS X by booting from the Recovery HD using the Command and R keys. When the Utility Menu appears select Reinstall OS X then click on the Continue button.
9. Erase and Install OS X Restart the computer. Immediately after the chime hold down the CommandandRkeys until the Apple logo appears. When the Utility Menu appears:

1. Select Disk Utility from the Utility Menu and click on Continue button.
2. When Disk Utility loads select the drive (out-dented entry) from the Device list.
3. Click on the Erase icon in Disk Utility's toolbar. A panel will drop down.
4. Set the Format type to Mac OS Extended (Journaled.)
5. Click on the Apply button, then wait for the Done button to activate and click on it.
6. Quit Disk Utility and return to the Utility Menu.
7. Select Reinstall OS X and click on the Continue button.

In High Sierra, only root has unfettered access to the root of the hard drive. You cannot give yourself permission. Just because the root of the hard drive is protected doesn't mean you do not have permission to subfolders. So, the issue with the permissions on the root of the hard drive are not the issue with your other problems.

Not being able to open applications is odd.

Permissions outside of the Home folder can be repaired during installation, so reinstalling may be the only way to fix the issues.

Having issues with things in your home folder is separate issue. You can repair those following this: Resolve issues caused by changing the permissions of items in your home folder - Apple Support

There is no problem with the Main SSD on my Mac Pro permissions - I can read & write.

What I was trying to explain was that the permissions on your Main SSD are wrong. They were carried over from what you had before the upgrade.

Normal permissions for any hard drive in High Sierra is System (root) read & write, Wheel group read only, and everyone (others) read only. SIP prevents anyone from changing those permissions. The only user that can write to the root of the hard drive is root. But, even root cannot change the permissions.

I have no idea if this was intended. Seems to be a rational security measure and they haven't altered it in three versions of the OS, now.

If you want to change you other drive's permissions, you'll have to disable SIP, change the permissions, and re-enable SIP.

I have the same problem but can't boot from another volume with my MacBook Air since I have only a USB means of connection in my remote location, and it fails to boot that way (even though System preferences / Startup Disk suggests it can by accepting selection of the volume).

"You had the command to turn it on. Use disable to turn it off.": I have no clue what you mean by that - it's much too cryptic to be discernible. Please elaborate with sufficient detail to follow...

In the meantime why does Apple saddle us with so many significant problems with most upgrades in this era? Boot permission corruption isn't the only one this time - all my folders lost their location and size information too, so every last one of them had to be reconfigured from scratch, a massive waste of time. Users don't have time to repair extensive damage from abject software design thoughtlessness...

The Mac OS was a far more user friendly environment in earlier times...

These are not hard drives - they are solid state storage volumes, or just volumes for shorthand. Accurate terminology matters...

To be "accurate," a Volume is a mounted share in the OS or a sub-section of an APFS Container.

As to the permissions, it has nothing to do with SSD or HDD or BFD.

"You had the command to turn it on. Use disable to turn it off.": I have no clue what you mean by that - it's much too cryptic to be discernible. Please elaborate with sufficient detail to follow...

The OP listed the command to enable SIP. The only difference between enabling and disabling is to replace the word "enable" with "disable."

In the meantime why does Apple saddle us with so many significant problems with most upgrades in this era?

It should have never mattered to anyone as you shouldn't use the root of the hard drive for anything.

So there. I would be very interested in what you discover. I, too, have this problem on one machine but not others. SSD in my MBP - no problem. SSD running iMac- similar problem to yours.

The upgrade didn't alter your Main SSD permissions. The default is only the bottom three on the list and you cannot edit the list of users/groups.