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I believe you can code in xamarin (for free) using visual studio 2015

With Xamarin, you can use it with both Xamarin Studio and Visual Studio. I believe XS is free now and you can get VS Community for free too.

But not under Linux. XS is available only for Mac, and VS is Windows-only (of course).

Of course, you can use it. Tools are under products at http://open.xamarin.com/

Microsoft has not open sourced Visual studio (VS) so similarly they will not open source Xamarin. You can also use .net framework without using VS. Just use command line tools

You don't need to use an IDE in order to write Xamarin.iOS or Android apps. You can just use a text editor (although Xamarin Studio is free).

The MSBuild logic, at least for Xamarin.Mac and Xamarin.iOS has been made completely open source, so all the logic to build your apps + the SDKs are all open source.