Hi!
To use the plugin with Elementor, simply use the search shortcode within any content element that supports shortcodes, and that is it. You can also use the search widget, as Elementor supports that as well.
The widget you are using there is not the search bar itself, but the top searches widget.
1. The search bar is individual, but you might be able to recreate that design on the plugin option, under the Theme Options panel. You can choose a theme, then customize it as you like – search bar colors, magnifier, search button etc..
2. You can have it active on the whole website, you can either use the automatic replacement feature, that works for search bars integrated to the theme, or use the search widget on a widgetized area that is visible on the whole site.