Got some help from Ralph today and now the JEditorPane is as I want it, e.g., code folds are now supported once you click in the JEditorPane, though there are still some side effects, since this is not how anyone anticipated NetBeans editor APIs being used. But, so far, the side effects (e.g., now the hyperlinks work, but they open a new JavaScript file when you click on one of them, instead of jumping within the JEditorPane itself) are not so terrible.
Error checking is also done now, which wasn't there before, i.e., red underlines and error annotations in the right margin. And maybe it's my imagination, but the editor feels a lot snappier, e.g., in code completion, than before.
I've checked in the changes, they're all in this file: