Is it possible to write an android app using C#? Is there an API or something? Is it free for personal and commercial development?
What do I need to install to have android project template in my Visual Studio?
Having used Mono, I would NOT recommend it. The Mono runtime is bundled with your app, so your apk ends up being bloated at more than 6MB. A better programming solution for C# would be dot42. Both Mono and dot42 are licensed products.
Personally, I would recommend using Java with the IntelliJ IDEA dev environment. I say this for 3 reasons:
I have been a C# programmer for 12 years and started developing for Android with C# but ended up jumping ship and going the Java route. The languages are so similar you really won't notice much of a learning curve.
P.S. If you want to use LINQ, serialization and other handy features that are native to C# then you just need to look for the equivalent java library.