Some evil recursive tree wrestling later, we can now find out the names, line and column numbers of all functions in a JS document. I also added some logic, to look up the tree for anonymous functions to see whether they were declared in an object literal, so we can use the key as the inferred name.
The parse tree could be used for all kinds of interesting things, like outlines and even IntelliSense(TM)(R). JS being a very dynamic language this, however, only goes so far. In particular systems like Joose or even dojo's simplistic object system do not go so well with static analysis.
As a next step, I'll move the parser to a web worker (I designed the API to be async, so this should be easy) and implement a simple outline view.