ASP.Net MVC Website & Web API As Single Implementation

I have a client dashboard application, written in ASP.Net MVC 4 (not bound to this version, happy to upgrade). The website is hosted in Windows Azure.

Our clients would like to programmatically access this dashboard, via an API.

Affectively they would like to be able to perform all of the same functions which they normally carry out on the dashboard website, programmatically from code using an HTTP Restful Service.

....My instant reaction was to simply build an ASP.net Web API project, and separate out the shared services/libraries/components from the existing MVC project so that both the API and the MVC website can call the same code base.

Questions

1. Is it possible to simply create Web API controllers within my existing MVC website project, and expose them over HTTP?
2. If it is not possible to do "1.", will Azure play nicely if I have an MVC solution, a separate Web API solution, and a shared library project of common services and models? How will I ensure that Azure copies the shared library components into the cloud when I deploy the MVC solution and the Web API solution separately?

Update

Based on the comments, the following implementation of two separate controllers (the original MVC controller within the MVC project, and an additional Web API controller within the same project), does in fact work. Please note that based on the following LINKED ARTICLE, the below implementation would be a "pre-MVC 6" implementation. MVC 6 itself provides the ability to implement both API calls and normally MVC View calls, into a single controller (as opposed to separate controllers that inherit from different base classes).

public class ProductsController : Controller
{
    //Products/Index
    public ActionResult Index()
    {
        return View();
    }
}

public class ProductsAPIController : ApiController
{
    // GET api/productsapi/getall
    public IEnumerable<string> GetAll()
    {
        return new string[] { "value1", "value2" };
    }
}