I’ve reseleted my menu as Primary. This solved the 2 x Home, but my menu has become vertical again, with bullets and underlines and >> indicating subs. The order and structure is fine and the drop downs act as fly-outs.
All is correct if I don’t use a child theme and use custom css with exactly the same css code, but I feel now I’m not far away from a solution and would really like to understand the styling issue and how to use a child theme correctly. Would it help if I emailed you the site url/login? It’s not ready for public yet.