How do I get Web API to render a Razor view using the model that it returns? And only XML/JSON when the accept headers (or .extension) is set? Is this even possible?
It
Yes, it is how it is designed for: Web API for data and MVC for rendered views. I know that some people will try adding view engine support to web API but it is not designed for it.
My personal view on this is that this parallel world between MVC and Web API (which is the source of most criticisms while community has generally praised the product) is mainly a consequence of the fact that Web API has been added to MVC without having a reference (or knowledge of it).
As Jon Galloway said on a recent podcast, had the team have HTTP knowledge they have now (as well as hindesight of the popularity of REST API now which they did not have then), they would have designed just a single pipeline serving data and rendered view alike.
I can only speculate that the future version of MVC/Web API will be presented as a single pipeline. In fact, this parallel world might have been a careful plan to unify them in the near future.