In what file I should include my code if I want content to come up on top of everything?

If you want to echo something extremely early, you can do that at template_redirect hook that runs before any template is processed. However that will effectively put output outside of proper markup and so markup produced will be invalid.

I do not quite understand why you need to actualyl output it early, if presentation is controlled with JavaScript (I assume?). You could as well echo it at end of page hidden and position/show when and how is needed on button pressed.