Hi Mads,
To restrict items to a specific category only, check this documentation for a step-by-step guide. It should do the trick 🙂
There is no pop-up feature, however there is a compact (slide out) search box feature. However most themes or page builders nowadays have a pop-up feature built-in, if yours have that feature too, then I recommend using that and placing the search box to the pop-up itself.