Then there's no way I can diagnose/fix the issue.
FYI, there's no much risk as the tweaker doesn't touch system files, only it's own in the installation path.

It works fine here on Windows 7.
http://i.imgur.com/ZpmhP6J.png

I can't think of a reason for it to fail, as the changes from v4.1.5 are really minor.

Can you try to update manually, by downloading the setup from the website?

Fixed in v4.1.6.

Do you get any error message? Or is it just loading forever?
You might want to try Softpedia:
http://www.softpedia.com/get/Tweak/System-Tweak/7-Taskbar-Tweaker.shtml

Here's a test build which does what you requested:
https://dl.dropbox.com/s/mfs3gwwjtu6k8su/7ttt.rar?token_hash=AAHInjPXiF14YfF5Zat8ZIZcwwTmwbP9RWasCvpAYGkYuw&dl=1

You might want to try it out, and see how inconvenient it is (in my opinion, at least).

I don't know how it works in Linux, but here, it's either you click early and get the list, or you don't click early enough and have the thumbnails shown.

There's an add-on for Firefox which sets different App IDs for different profiles.
http://rammichael.com/7-taskbar-tweaker/comment-page-9#comment-1872

There's no registry key for this. It's a Windows bug, and I'm not sure a solution exists.
In general, it happens because Windows replaces a hanged window with another, pseudo Window, with a "Ghost" class. This replacement is synchronized with the taskbar. But when the window gets responsive again, and replaces the "Ghost" window, the replacement isn't always synchronized, and sometimes the taskbar thinks it's a new window.

Looks like it's not the tweaker's fault after all.
I've never encountered such an issue, and I can't think of a reason for the tweaker to screw up thumbnails, especially with the default options set.

Can you reproduce it, or is it random?
You can turn off the tweaked and see if the issue shows up.