Could not back up the Phone “iPhone” because not enough free space is available on this Mac.

OK, me again. I figured this out. AFAIK, my iPhone's OS was damaged so I had various problems, see above for short descriptions.

The solution was to offload a lot of apps (so I had 3GB free on the phone), reboot the iPhone, connect to the Mac, and update to iOS 13.6.1 using the Finder.

https://support.apple.com/en-au/guide/mac-help/mchl0e5e3d56/mac

This process asked if I wanted to backup the phone before installation, so I say "Yes" even though the backup, by itself, failed earlier. This time it worked.

NOTE: What I do not know is whether or not this would work if there was no update available. For instance, would I have been able to do a backup if I had offloaded the apps and then tried a Finder backup, and then a Restore? Would this have fixed the problems or would it have just reinstalled my faulty OS? Why did the original backups fail on both a Mac and a Win 8.1 PC, both with plenty of free space? Why did the icons disappear? Why the memory fluctuations? Why, why, why? Akk.

Oh, well, fixed for now, so far, so good.

Further notes: I don't want to wipe it out and start over since I have a few high scores I'd like to keep for a couple of games. I did sync via iCloud my contacts BUT have not been able to sync Notes (I sent them to the Mac by AirDrop).

Rebooting didn't help. Changing cables didn't help. I can't update to 13.6.1, if I click on Settings>General>Software Update>Download and Install I get a low memory warning "Temporarily Remove Apps to Install Software Update?" appears. Click on "Continue" and the warning goes away, some apps seem to get offloaded, but nothing else happens. The iPhone has 1.3GB free though this will drop down to under 200MB for seemingly no reason. And then go back up. Sometimes a reboot is necessary. Sometimes not.

I was having small problems with "not enough memory" under iOS 13.5, but these problems have sprung up under 13.6. So far, my wife's phone has not had these problems.

As of 8/25/20, still working fine. So offloading lots of apps was the solution. And, of course, keeping regular backups!