The easiest solution is of course to see of JavaFX is supported on Raspbian. If it is, then you can use the WebView component (http://docs.oracle.com/javafx/2/webview/jfxpub-webview.htm) and most of your problems will be solved. As pointed out in the comments below by jewelsea, this is not a feasible solution yet. This might change with future builds of JavaFX for Raspbian though, so keep an eye on it.
If, for some reason, JavaFX won't work (due to display drivers etc.) then you might want to reconsider your application approach. If you just need to display the HTML file, then you can write it out to disk and display it in the default system browser. If you need to interact with it, it becomes a bit more complicated - you'd basically write a small web server and then respond to user interaction that way. The user will be using their browser of choice, and you'll send responses back to it.