Is there a Menu reference module I can use to attach menus to nodes, and render them on the page?
I have never used the module, but in theory, Block Reference should be able to solve your problem. If needed, I assume you can combine it with Menu Block to set up the menu blocks you need.
You can control which pages to display blocks (menus) on. It would be quite simple only to show a specific menu in a page and region of your liking.
This is done in the block section.
Yes there is a Menu Reference Module
Consider the Context module. Awesomeness packaged.