web api: Route different from name of controller, possible?

If you are using Web Api 2 you can use Attribute Routing for this:

[RoutePrefix("api/mine")]
public class mineController : ApiController
{
    [Route("method1")]
    [HttpGet]
    public IHttpActionResult Method1()
    {
        //Route would be api/mine/method1
    }

    [Route("method2")]
    [HttpGet]
    public IHttpActionResult Method2()
    {
        //Route would be api/mine/method2
    }
}

Yes, by using Attribute Routing.

Step 1: Enable attribute routing in WebApiConfig.Register method (might be on by default, I don't recall offhand):

config.MapHttpAttributeRoutes();

Step 2: Not required, but it's nice to use a RoutePrefix attribute for the entire controller:

[RoutePrefix("api/mycontroller")
public class mineController : ApiController
{
    ..
}

Step 3: Use a Route attribute on each method that completes the route prefix:

[Route("myaction")]
[HttpGet] /* or other HttpVerb */
public IHttpActionResult SomeMethod()
{
   ...
}

[Route("myaction/{id}")]
[HttpGet] /* or other HttpVerb */
public IHttpActionResult SomeMethod(int id)
{
   ...
}

More info here: https://www.asp.net/web-api/overview/web-api-routing-and-actions/attribute-routing-in-web-api-2

You can use the System.Web.Http.RouteAttribute to decorate your controller actions and specify any route you desire. Depending on what it is you want to do, this may be a good approach for you.

There is also the option of using a ~ to bypass the route prefix

take this from the ms documentation

[RoutePrefix("api/books")]
public class BooksController : ApiController
{
    // GET /api/authors/1/books
    [Route("~/api/authors/{authorId:int}/books")]
    public IEnumerable<Book> GetByAuthor(int authorId) { ... }

    // ...
}