Is there a way by which I can use Git not only as a code repository but also for a build and continuous deployment to Azure (Cloud service, NOT web site)? I suspect not given th
There is support in the new management portal for directly publishing "Web Sites" from Git, but not for Cloud Services as you have already noticed. However, there is support for doing that with TFS. There is currently a free TFS hosted preview available, and I have a blog post that walks through configuring Continuous Integration with Azure Web Sites and TFS Preview. The process is nearly identical for a Cloud Service.
Update : This is now possible
http://www.windowsazure.com/en-us/documentation/articles/cloud-services-continuous-delivery-use-vso/#step6
Working like charm with both git and tfs.