Regarding the port 4002: this endpoint is hopefully going to provide a basic admin UI, but it’s not been implemented yet. There’s a basic diagnostic dashboard at host:4002/dashboard (it’s this one GitHub - phoenixframework/phoenix_live_dashboard: Realtime dashboard with metrics, request logging, plus storage, OS and VM insights), but for the most part you won’t find it too useful unless you’d be debugging performance/resource consumption issues.
In other words, at the moment there’s not much there and exposing it in your proxy is not essential, but not harmful either (as long as you don’t expose it to the public internet, as it doesn’t do any auth).
Re custom CSS: have you checked this Customization - asciinema docs ?