Again, thank you for your prompt attention to this matter.

When I went to your link, I just saw a screenshot, not a video, but that didn't matter.  I made my settings match yours (I presume the default) and the bug did NOT manifest itself.  Here is a list of the differences between the settings shown there and the ones I used (which DO manifest the bug):

y drag to reorder

n group by App ID
y don't group
y when not grouping..
y don't group pinned items

n use taskbar default settings
y don't combine grouped buttons

y cycle between taskbar buttons

y display seconds on the tray clock
y reserve empty space on taskbar

To switch user, I go to the Start menu and choose it from the various Shutdown options.  I have three users defined on my machine.  I don't know if "switch user" is available if there is only one user.

I tried Win+L once and it did NOT rearrange the buttons!

The issue is not specific to Firefox, at least in terms of buttons from different groups changing at least group position.  I noticed that buttons within Chrome stay in the same order.

Thanks for looking at this issue.

Thanks for your attention to my comment.

I tried the following:

1. exit 7+ Taskbar Tweaker
2. Switch User
3. log back in

The
task bar buttons are in the same order, at least within the same
application.  in my case, I have about 20 Firefox windows open.  I use
the Firetitle extension to name each window and then I alphabetize the
taskbar buttons.  They stayed in that order when 7_ was NOT running, but
not when it was running (though they don't always reorder that much).

BTW,
I love your program.  I wouldn't consider using Windows 10 without it! 
This is a relatively minor issue, though I have recently started
"switching users" when I leave my PC (for privacy), so it is becoming a
bigger issue.

To answer your questions:

Yes, multirow taskbar.

When Tweaker is NOT running, I can't move any of the taskbar buttons.

When I uncheck all of the Tweaker options, I get the same symptoms (no insertion point)

As I said, I can live without the insertion pont, as what is most important is moving the buttons around . BTW, I typically have well over 10 Firefox windows open and I like to arrange them alphabetically by name. I can name each window using the FireTitle extension.

These are the settings I typically use. The attachment didn't work before because I had the image in a Wordpad file.

I am running Windows 10 Professional (64-bit) (with anniversary update)

I am using a 4-row taskbar, but I don't think the number of rows makes a difference.

image of settings is attached