In at least some of the extensions, the dialog window is pushed underneath the main window when the user clicks on some other part of the UI. If the user wants to use the extension repeatedly, it is frustrating to have to bring the window to the foreground over and over. Is there some way to have the extension dialog stay visible until dismissed?

Can you give an example of the extensions and OS which do this ? 

For a standard Inkscape extension which uses the extension gui Inkscape provides, i.e. the gui elements are defined by an .inx file, then this should not happen.

If the extension uses an gui separate to Inkscape ( python Tkinter or Gtk3 ) this is mostly unavoidable. 

The Jitter nodes extension is an example of the behavior I mentioned.

I did the following test.

1) Draw a line using Draw Bezier curves

2) Open the Jitter nodes extension.

3) Click on Apply.

4) Click on the main windows anywhere outside the extension window, and the extension window disappears.

ASUS VivoBook laptop

Windows 10

inkscape 1.1

Same on macOS. That´s why I disabled the Welcome-screen because when I open a file while Inkscape isn´t running in first place the Welcome-screen is hiding behind the main window and you can´t do anything until you find it and close it. Learned it the hard way.

I've booted into my windows 10 machine, with Inkscape 1.1, you are correct. I didn't realise this

The behaviour I described is from my Ubuntu machine. 

Yes.  I'm on Windows 10.  I often have several windows up, so it's a pain to find the dialog.