Additionally, while on a wide screen the tools are on the top right - a very prominent place - on a small screen they are bottom left - probably the least prominent place. This is a bit strange, too, especially for users like me who have the window width just at the threshold, and sometimes increase and decrease the width.

I think a dropdown menu from the header tools makes the most sense, just to the right of "History." The right floating version takes up too much real estate on the screen, and when it pushes below the left it's basically hidden entirely.