If the menu with the items is hard coded, you will likely have to use the php shortcode to place it to the desired position. It is possible to use the search shortcode as a menu element as well.
You can enable the compact box layout for the pop-out search bar under the Layout Options -> Compact box layout panel, that is probably what you are looking for.
The live search results can be moved on the current page using the results shortcode. The regular search results layout is controlled by the theme, within the search.php file in the theme directory. It cannot be moved, as WordPress core automatically resolves to that page.
I hope this helps!Best,
If you like my products, don't forget to rate them on codecanyon :)