1. If you use the option to show search over header image / theme slider / rev slider from theme options – help screenshot, the option applies for all the pages with such headers.
2. If you select to show or not advanced search, it is for all headers
3. If you wish to not show search in a specific page, but still use the global option to show search over header with image, theme slider or rev slider, you can use Header Options from the edit page and select Header “none”.