I have only been able to make the File Browser completely break on some phones when switching between portrait and landscape view. When starting in landscape view, sometimes the File Browser only shows the description and the submit button (no images).
In most other phones, the overlay never takes up the full width, even though it is supposed to when the width is narrow. I have not been able to make the File Browser extend off the screen horizontally without switching to landscape, and only then it only extends off the right side of the screen.
Regarding the scrolling problem, I'm not sure yet what we can do about that since it's controlled by the browser, but I can understand why it might be undesirable to accidentally scroll the parent window while an overlay is still open.
If the plus-icons are difficult to use to choose images on mobile, I can imagine how hard other things might be: like the Book tab, or trying to remove a chapter in the editor.
I will try to look at all of this next time we're working on style issues. This week the focus is on race conditions and certain situations that cause data loss.
If you know specific phones that are having issues with File Browser, please let me know.