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I see that UWP is in the .NET Core box. Does this mean that we can or will be able to write GUI apps with UWP that target Linux?

No. Think of UWP as WinRT + .NET Core. While .NET Core is available for Linux, WinRT isn't. Within UWP the UI stack is provided by the WinRT side.

Xamarin has an abstraction layer for UI called Xamarin Forms. This will allow you share UI code across .NET Framework, UWP, iOS, and Android.