To answer your queries..
- Converting your existing web api to azure mobile services
Have a look at this blog.. http://blogs.msdn.com/b/azuremobile/archive/2014/04/10/creating-an-azure-mobile-services-net-backend-from-scratch.aspx
It talks about developing azure mobile service from scratch. It has the required bits you need to modify your existing web api project to a mobile service.
2, Yes, this is possible..
See this blog I wrote http://www.supreet-tare.com/2014/04/create-multi-table-dot-net-backend_24.html
Or see this one that details how to configure mobile service web api project to run locally
Hope this helps. Supreet