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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.

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).

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

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