I cannot figure out how to add the search form to a child element in the menu!
The search button on that site looks like it is a menu item however you will find that it is a separate entity, nothing to do with the menu at all.

Usually that kind of thing is achieved using Javascript and AJAX.  Javascript to make the form appear, and AJAX to submit it.

Have a look at the Popup module, it might be what you are after.