For example, if I have a friend over and he wants to show me a video using a given app that runs both on my device and his device. Could that app display a QR code on the screen or something that he could scan and instantly be granted access to my Chromecast device?
As Ali mentioned, Chromecast devices are discovered and apps launched via local network applications. One an app is started, it could easily connect with a cloud service that allows other (non-local) devices to talk w/ your Chromecast via the cloud service. A Chromecast Receiver application is just a HTML5 application (HTML, CSS, and JavaScript). You can really do what ever you want once your application gets launched.
If displaying a QR code that allows some kind of rendezvous with your cloud application is what you want to do, you can certainly do that.