It's possible for some extent, see the following answer:

https://tweaker.userecho.com/communities/1/topics/287-ability-to-run-application-or-even-better-trigger-a-media-button-on-clicking-on-empty-taskbar-space

Thank you for the suggestion. This was already suggested:

https://tweaker.userecho.com/communities/1/topics/318-7-taskbar-tweaker-settings-in-context-menu

Closing as a duplicate.

Sounds like this issue:

https://tweaker.userecho.com/communities/1/topics/592-taskbar-inspector-not-remembering-path-with-invalid-chars

Closing as a duplicate. If you meant something else, please leave a comment and I'll reopen it.

I replied to your email.

As far as I know, you're wrong - hovering over the start button doesn't open the start menu, not after a short delay and not after a long one. I don't know why it happens on your computer. Either you click on the button accidentally, or you have some other customization software which adds this functionality, or perhaps you enabled some accessibility feature.

I'm not sure I can help you with this unless you can provide more details that explains this behavior on your computer.

The website served 5.5.0.6 because the file was cached. It should be 5.5.0.7 now, please try again.

Please try 7+ Taskbar Tweaker v5.5.0.7 beta which I just released. It should fix both issues.

You can drag Taskbar Inspector from the gray are and move it to the desired monitor, then it'll display that monitor's items.

Anyway, your screenshot indicates that there's an issue - the gray area shouldn't be that big on the bottom and on the right. I'll check it when I have time for it.

The context menu problem is a result of the following issue:

Dialog does not support DPI scaling

You can try forcing DPI on 7+ Taskbar Tweaker. Here's a guide on how to do it.

The Taskbar Inspector is a completely different issue. It's not related to DPI, since Taskbar Inspector runs from the context of the explorer process, which supports DPI. I've never experienced this issue - can you try to provide a way to reproduce it? Does it happen on every monitor, or only on one/some of them? What happens if you resize it?