I want to change the order in which menus and programs are presented in the panel menu, so right-clicked to get to the menu editor. A first problem is that when I want to move a menu anywhere but to the top, it ends up as a sub-menu under another menu. I thought I fixed that, but now the system menu is under the internet menu and the graphics menu is under that. No matter how often I change that in the menu editor, it remains the same. I have done that so often that now I have a graphics-5 and a system-7 menu (both empty) in the menu editor, which however do not show up in the menu. What is going on here?

If a menu is empty in the menu editor, then it is automatically hidden by the "Panel menu" ( Kickoff ).

It sounds also like your menu has become inconsistent, so please run "kbuildsycoca4 --noincremental" to ensure that KDE's cache of the menu is up to date and consistent.

I don't use kickoff but classic menu style, but hiding empty menus makes so much sense that it also has to have that feature. So it was stupid of me not to think of that.
I ran kbuildsycoca4 --noincremental. That took some time - a long list of messages, many of which of 2 types:
- something in usr/share/applications is not compliant with XDG standard
- something in usr/share/applications specifies undefined mimetype/servicetype
Tried again. No, didn't help. When I make changes in the panel menu editor I get the message "updating system configuration", but the settings don't 'stick'. The order hasn't changed and I still have System under Internet and Graphics under System. Even though I removed all the numbered empty menus I now have System-8 and Multimedia-3 and such.
Hold on, it does work if I make a new System menu and copy something to it using copy and paste instead of dragging. But that is going to be a hell of a job, copying it all, since I don't see how I can move a whole sub-sub-menu up.

If you wish to revert the changes you have done to the menu, simply remove ~/.config/menus/applications-kmenuedit.menu and run "kbuildsycoca4 --noincremental" again.