Yeah I meant to mention it too - the mobile menu closes when you click on the search, presumably because it's set to do so onblur.
I just hack it by going to the search page address and adding my search query to the URL:
http://www.jehovahs-witness.com/search/?q=
(Also an empty search gives a "Server Error" message rather than a slightly nicer "You didn't search for anything!" message.)