#1
Yes, You can create the mobile responsive layout of your header using the plugin.
The Elementor Header & Footer Builder plugin just gives you a container where you can completely design the header using Elementor Page Builder, So the process of creating the mobile responsive layout is exactly the same as you would create a responsive layout of your other Elementor page.
Here is a documentation by Elementor Page builder which explains how you can create mobile responsive layouts using Elementor – https://elementor.com/introducing-mobile-editing/
This same applies when you are creating your Header/Footer using this plugin.