Is DOTNET provides a common API to all languages ?

Yes, that was one of .NET's core design goals. Microsoft developed an open spec called Common Language Infrastructure (CLI) and the .NET framework (in particular the CLR) is a specific implementation of the CLI. Mono is also a CLI implementation (from a different vendor). The core idea is that all CLI languages can use a common API and can easily interop with each other. Practically speaking, not all languages implement all CLI features, so eventually there are exceptions where language interop may not be smooth. To work around this, there is a sub set of the CLI called the CLS - Common Language Specification. CLS compliant languages must implement the CLS (plus any other CLI features they want to). As long as a CLS compliant language uses only CLS complaint features, then it can smoothly interop with any other CLS compliant language. VB.NET and C# are both CLS compliant languages.

Yes of course. For all .NET languages.

—SA

If by all you mean non-.Net supported languages, then no. They will have to adhere to their capabilities to use something that is written in a .Net dll.

If your do not care if it is API or anything else, then yes: webservice/WCF service.

If you mean something else, explain it.