Currently, having a "hide when not in use" option would be near impossible to do without severely impacting performance. If something changes in a future version of Firefox that makes this option more doable, I'd love to add it, since you aren't the first person to have suggested this.

As for the issue that you are having, when I test on my computer, nothing about the drop-down menu on pages change for me. This isn't unexpected, since the drop-down selection elements on pages are treated as part of the Firefox UI, which cannot currently be modified by add-ons.

However, if you are getting some other behaviour or something is breaking on the page, please feel free to email me at addons@wesleybranton.com with a link to the page you are having issues with, the settings that you are using, what operating system you are using on your computer (Windows, Mac, etc) and a screenshot of the issue.