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.

Looks like the item that you've disabled in Task Manager was saved in a different place, other than "HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Run".
I don't think Task Manager provides a way to see the location. You can enable/disable the entry, and use a program such as Process Monitor to see what is being changed.
http://superuser.com/questions/79566/whats-the-eas...

I'm using the built-in Administrator account. Could that be the problem?

I don't think so. The setup is quite common, and yet you're the first one to report about such an issue.

After disabling it in Task Manager, does it still appear in Autoruns?
Is the checkbox in the tweaker's settings checked?
Have you checked the following branch?
HKEY_CURRENT_USER\Software\Wow6432Node\Microsoft\Windows\CurrentVersion\Run

Yes, that's OK.
As I assumed, the tweaker runs twice. The second time it causes the main window to be shown.
Perhaps the Autoruns program can help you here: http://technet.microsoft.com/en-us/sysinternals/bb...
See if you can find a second instance of the tweaker.