Your comments

You are the first to report about this kind of issue. I can't do much with it without a way to reproduce it.
Also, why do you think it's caused by the tweaker, and not by another program?
That means that when inspector comes up, it covers the taskbar, but since the taskbar itself is redundant at that point, it doesn't matter.
Yep, that's how it worked before. I've changed it due to a request of one of the users. Overlapping the taskbar indeed seemed less correct.

Is the box only there if focus follows mouse, or did you add it for all situations?
It's there only if the "focus follows mouse" option is enabled.
Windows allows you to choose the monitor to show the clock on, by choosing which one is the main monitor.
As for displaying the clock on multiple monitors, there's an existing suggestion about it, please vote if you're interested:
http://tweaker.userecho.com/topic/378660-add-clock...
(Maybe I should try installing an older version to see if I can confirm what I am claiming, but I don't know where to find older versions, since installing the new one clobbered the old one.)
Yes, this is interesting. You can grab old versions of the tweaker here:
https://www.dropbox.com/sh/mcbychsh0xc1dvu/8-Lj2sg...

The trouble with checking for the focus-follows-mouse setting is that if MS changes how they store that, the fix will no longer work, but if you want to do it that way, I have no problem with that.
This is documented, and shouldn't change in the future.
SPI_GETACTIVEWINDOWTRACKING

Also, I assume you mean click on desktop, not on a blank area of inspector. The latter would be bad, imho, since clicking on something usually selects it, not closes it.
Actually, I meant the inspector window, not the desktop. And that's why I talked about the right click.

As yet another option, what you might want to do is add yet another checkbox to the tweaker menu to toggle exit by focus vs exit by click.
I don't think there's a need for an option - detecting the accessibility option should work. The question is - what kind of click should trigger exiting?

And, yet another option is instead of checking whether focus follows mouse is set, you might instead want to check the menu delay
I'm not sure that it's related, and even if it is, I prefer not to change it with the tweaker.

Anyway, you're probably the only tweaker user who has this accessibility feature turned on, so think about a preferred exiting method (which is not difficult to implement :)) that you'd like to have.
I could reproduce the issue. Indeed, Taskbar Inspector seems to not be compatible with the "focus follows mouse" option.
What solution would you like to see? I prefer to avoid adding buttons to the inspector window. On the other hand, there has to be some option to close it with the mouse.

How about this: the tweaker will check for the conflicting option, and if it's turned on, it will not close upon focus loss, but it will if you right click on an empty space of the list, or the border. What do you think?

P.S. currently, you can hold Ctrl+Shift while opening the inspector to keep it open. Perhaps you'll find it useful.
I could reproduce the issue. It will be fixed in the next version.
I can't reproduce it.
What is your Windows version? 32/64-bit? What is the version of 7+ Taskbar Tweaker?
Also, posting the tweaker's options and the taskbar's properties might help reproduce the issue.
If you're using the portable version, then the last_taskbar_width value should reside in the .ini file, not in the registry. How come that you're seeing it in the registry? Did you have the non-portable version installed before?
I understand. I guess that it's a matter of preference - some will find it more convenient, some won't.
The tweaker doesn't provide this kind of fine-tuning, it just uses the system volume control. If you want more customization, perhaps using a specialized tool such as Volumouse is the way to go.