Will the index.html swf need to be re-compiled in order to embed the flexviewer app in between a html header and footer wrapper? Or it this perhaps handled in the HeaderControllerWidget.mxml?
To answer your two questions:
No, it doesn't have to be re-compiled.
No, it's not handled in HeaderController.
Editing the HTML file is a good idea (like Brad mentioned in his answer). We showed an example of that at the Esri Developer Summit a few months ago. Just take care to edit both default.htm and index.html (or know which one your web server defaults to).