How to add a search input field beside an usual browser input cell?

You can drag the search bar out of the Customize palette to the Navigation Toolbar to get a dedicated search bar. *https://support.mozilla.org/en-US/kb/customize-firefox-controls-buttons-and-toolbars If you want to adjust the width of this search bar then you need to place it directly adjacent to the address bar and in normal mode place the mouse pointer between the two bars and drag their relative width to the desired widths.