Fixed in v4.5.2.

Should be fixed in v4.5.3.
The issue existed since v4.5.1.8 beta (September 21, 2014), but unfortunately nobody has reported the bug until the final version was released.

Yes, the "Don't group pinned items" option has a couple of side effects. Additional example:
http://tweaker.userecho.com/topic/314270

Currently I don't plan to fix those, as I believe that they are not serious enough compared to the amount of required work to find a solution.

v4.5.1.8 beta fixes most of those issues.
The only remaining issue I'm aware of is that Taskbar Inspector doesn't show an icon, and sometimes doesn't show a correct title.

A duplicate of this, please see my reply there.
tl;dr should be fixed in v4.5.1.8 beta.

Perhaps X-Mouse Button Control can help you here.

Предложение уже существует:
http://tweaker.userecho.com/topic/173756-option-to...

Well, that was only a guess.
Seems like some kind of a Windows glitch, which was probably sorted by removing and recreating the startup entry (which is what the checkbox in settings does).
I'm glad that it works as expected now.

Try the following:
Re-enable the item in Task Manager, and uncheck the Run at Startup option in settings.
After a restart, the tweaker should not launch automatically on a typical system, but I think that it will launch on yours.