You can use WebSharper and get it with F#. It even has commercial backing. Comes with a neat little RPC system so you can effortlessly decide which code runs where.
There were also other projects that allowed this with C#, I just forget their names.
I don't think so. It failed with Silverlight, and was nasty with ASP.NET webforms. Microsoft now champions ASP.NET MVC on the server and TypeScript on the client. They also use TypeScript to write huge JavaScript apps for consumers.