There is a search bar feature on the desktop view but this disappears on mobile, is it possible to have a full row search bar on the top menu or the search icon on the bottom menu bar?
You can insert the Search element in the mobile view of the Heade Builder http://prntscr.com/iyd3v3 and configure it to be a search form only. Otherwise, the icon would call the mobile menu.
By the way, the mobile menu element has the option to show the search form in the top.