I fixed the button not displaying any text.
This is the new Javascript for it:
$(document).ready(function(){
var regionDropDown = $('.region_dropdown_section'),
regionButton = regionDropDown.find('button'),
regionList = regionDropDown.find('.region_dropdown_content').children();
$(regionList).on('click', function(e){
var region = e.target;
regionButton.text(region.text).val(region.text);
});
});
The only problem I still got now is to put the button on the right side inside the search bar, or on the right side of the search bar but against it.