Hi!
1. Positioning it relative is likely what you want. However your manu container is already in a relative position, I just checked, so that is already covered.
If you place the search icon to that container (ex. as a menu item), then set an absolute position of 0 to the right and auto to the left, then it should be positioned relative to that.
2. Those are the actual search filters. If they are set as visible by default, then using the shortcode generator or the setting module or shortcode you can poisition then anywhere away from the search bar on the same page. This setup is very similar to that and here is a step-by-step tutorial for it.