You should publicly post these usecases somewhere, too, as a type of "Request for Projects". It gives people ideas... after all, you spend more time thinking about your API than anyone else ever will.
Not necessarily. Sometimes what you want is to combine data from multiple sources in interesting ways. A single API doesn't need to be versatile, it just needs to make useful information available.
Many times, the only way to do what you want with data is to use the API to get everything, then transform it locally. This is obviously really inefficient (and sometimes impossible).
Disagree. The most important thing by far is to have an API at all. Just being able to build your main site but with a different UI (or as a native program) is a more than good enough use case to make it worth publishing an API.
By all means improve the API if you can, but having one at all is the most important thing. And as people start using it they'll naturally map out the use case space and you can see what cases come up - rather than trying to figure out which interfaces people will need up front, it's much more efficient to respond to feature requests as they arise.
"HAL is a simple format that gives a consistent and easy way to hyperlink between resources in your API. Adopting HAL will make your API explorable, and its documentation easily discoverable from within the API itself. In short, it will make your API easier to work with and therefore more attractive to client developers. APIs that adopt HAL can be easily served and consumed using open source libraries available for most major programming languages. It's also simple enough that you can just deal with it as you would any other JSON."
Plus it gives you a safe space to learn about the intricacies of running an API before you charge for it.
Also, as a developer if I heard "we are exposing an internal API and charging for it" I'd be confident that the API would be around.