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That's already possible with the advanced options. See my answer here, and adjust the value accordingly:
https://tweaker.userecho.com/topics/949-right-click-on-taskbar-items-to-minimize-window
So I understand that's it works not. If not, let me know.
The 7ttRefreshWnd window is used by 7+ Taskbar Tweaker to be able to apply settings without restarting explorer, while keeping the order of all taskbar items. It isn't supposed to be visible on the taskbar. I'm not sure why you're seeing it, perhaps it's a conflict with some other customization tool which causes this. If you find a way to reproduce it, please let me know.
The tweaker is installed in the %LocalAppData%\Programs folder by default. This way, administrator rights aren't required for the installation.
Do you mean that the tweaker doesn't start automatically, but works if you run it manually?
Make sure that the checkbox to run the tweaker at startup is enabled in the settings, and that the startup item is not disabled in Task Manager (the Startup tab).
Можете попробовать программу без иконки, см.:
I won't be adding it to 7+ Taskbar Tweaker, since it's not necessary to tweak the taskbar for that. There are programs which show the status of these buttons in tray icons. Here are a couple of examples:
https://www.softpedia.com/downloadTag/Num%20lock%20indicator
Sounds like more of a Punto Switcher bug than a 7+ Taskbar Tweaker issue. The implementation of the tweaker for double click handling is very simple and standard. I'm marking it as not a bug, let me know if you find a reason to assume otherwise.
The answer to your question is yes. In fact, choosing List under Hovering is similar to doing this registry modification (except the tweaker doesn't modify the registry):
https://winaero.com/taskbar-thumbnail-threshold-list-windows-10/
A relevant suggestion:
https://tweaker.userecho.com/topics/293-program-specific-thumbnaillist-hovering-option
I tried to implement the above, but gave up on it as it wasn't as simple as I hoped.
Customer support service by UserEcho
So you changed it to "Enabled" in Startup in Task Manager, and it still doesn't work? I'm not sure what else can be the reason, except perhaps a third party program blocking it. One of the users told me once that he had the same issue and he found out Avast was blocking it. If you're using Avast, or another non-default antivirus, you might want to check it.