Hi! Maybe I'm just talking to myself here, but after posting the above solution, I realised the GUI configuration buttons didn't work for any of the tray manipulations.
That is, clicking on: Menu -> Desktop -> Desktop Settings -> JWM Configuration -- then clicking on the "Tray management" button gives you the "Tray Configuration Options" sub-window. All 5 configuration buttons on this window appear to work, but when you hit the "OK" button on any of them, you get an error message. The "OK" button on the "Tray Configuration Options" sub-window works, but it doesn't save any changes.
The problem lies in the "/root/.jwm/jwmrc-personal" file. For some reason - possibly a result of that nasty "Or, apply current GTK theme to JWM" button, this file ends up with a few incorrect options. For the record, here are the 3 I had to comment out:
<!-- Number of virtual desktops -->
<Key mask="A" key="F1">root:1</Key>
NOTE: The first one may not be verbatim! I had long-deleted it by the time I posted this solution.
To find the errors stopping your GUI configuration tools from working, run this at the prompt:
Any errors requiring your attention should be revealed by this command. To fix them, use your favourite text editor (I use "Joe's Own Editor" at the terminal) and either delete the offending lines or comment them out. To comment out lines in these files, the format is "<!-- ... -->". For example, the last two lines from the above examples would look like this if they were commented out:
<!-- <Key key="F12">root:1</Key> -->
<!-- <Key mask="A" key="F1">root:1</Key> -->
Once you make these changes and save the file, restart JWM and the GUI configurations should work again. If you do find anything returns some kind of error, try changing the setting, saving it, then going back and changing it to its original. This is perfectly safe to do, as the changes only show through once you restart JWM.
At this stage, everything seems to be operational again. If anyone finds any other quirks, please post them here so we have some record of what the Puppy Linux dudes need to fix. I think I'll head off to the beach