It's not a bug of the tweaker, therefore I've changed it from a Bug to an Idea.

7+ Taskbar Tweaker v5.2 introduces the virtual_desktop_order_fix advanced option. More information can be found here.

I'm not familiar with the issue, but I found the following thread for you:

http://www.tenforums.com/browsers-email/20589-cant-unpin-microsoft-edge-taskbar.html

Perhaps the issue was fixed in v5.2. If not, please leave a comment or create a new topic.

I assume that it's not an issue anymore.

You should be able to remove it by right clicking on it and selecting "Unpin from taskbar". What happens when you right click on the Edge icon?

Try 7+ Taskbar Tweaker v5.2.1.

It was a false positive detection. Should be fixed now. VirusTotal page.

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)

That doesn't make sense. You should be able to reorder the taskbar buttons when the tweaker is not running - that's a feature of Windows. Also, having the tweaker running without enabled options should not have any effect.

In order to investigate the issue, I need a way to reproduce it. As of now, I couldn't do that with your tweaker settings.