You can't publish list as a page selector in NUX. It automatically takes the dimensions of the modules when you try to publish pages and make them Context Selectors in NUX. You have an option of hiding those context selectors though.
On Search functionality you will get that here as well. Just click on any context selectors and you will see "FIND" search bar.
By the way Context Selectors in NUX = Page Selectors in Classic