tray_icons_padding is supported in Windows 10 since 7+ Taskbar Tweaker v5.2, but it's implemented differently.

More details here.

У меня не получилось воспроизвести проблему.

На моем компьютере, 7+ Taskbar Tweaker качает новую версию отсюда, и корректно устанавливает.

Проверьте эту ссылку, какая версия качается с нее на Вашем компьютере?

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.