Can you please update your ticket with temporary back-end and FTP access details? The taxonomy and the styling problem might be caused by an incompatibility, but I'm not able to tell unless I see it.
The menu issue is most likely a simple z-index problem, whereas the menu elements appear behind the input field. I will most likely be able to suggest a custom CSS code to increase the menu z-index value to appear in front of the search bar – but of course I will have to see it first.Best,
If you like my products, don't forget to rate them on codecanyon :)