That is not browser toolbox - just normal devtools. Browser toolbox is separate tool which is used to inspect the browser window itself rather than web content. It’s essentially a separate Firefox instance with it’s own profile.

That is not possible. Browser toolbox runs in a completely separate Firefox instance in a separate profile so there’s no way you could display it inside the “main” browser window.

I’m guessing that the reason (and a good one at that) is that simply having an option to connect to a local chatbot leads to just confused users because they also need the actual chatbot running on their system. If you can set up that, then you can certainly toggle a simple switch in about:config to show the option.