We have a .NET Web Role hosted on Windows Azure that only serves a REST API with only a hand few web methods.
API is u
I've had a very pleasant experience with ServiceStack:
http://www.servicestack.net.
It's basically your last option; a fairly thin layer on top of HttpHandlers with fast XML and JSON serialization, that exposes a REST API.
The JSV serialization it also offers is about half the speed of Protobuf.NET I believe, and support for ProtoBuf is planned.
I don't know for sure if it runs on Azure, but I can't think of a reason why not as it simply integrates into any ASP.NET application.