1. In the newest version of GP (releasing this week) – there’s a filter that allows you to change the URL of the logo/header. However, you can also remove the header set in the Customizer, and add your own using GP Hooks in the “Before Header Content” hook. This allows you to add your own HTML as the header, so you can specify your own link.
2. The Spacing add-on allows you to make the navigation thinner – look for the “Element Spacing” area in the Customizer, and then adjust the Menu height.
3. You can use the Page Header add-on to add an image below your navigation.