Does Umbraco support SOAP or REST? or any other API?

Yes, it does. As umbraco supports .NET 100%, you can use any type of SOAP/Webservices you fancy.

umbraco also comes with “umbraco base” which is a proxy for calling .net classes via http (very handy for AJAX/JS) see:

• http://umbraco.org/documentation/books/introduction-to-base
• http://objects.dk/?filterby=/base