Hey Bhain7, thanks!
The Google Cast API supports two primary modes: media receiver and custom receiver. A media receiver app can use a stock Google provided app or it can customize the UI. These are centered around playing audio and video.
For games, you need to use a custom receiver with custom messages between the sender and receiver. A receiver is basically a Javascript app that runs on the Chromecast and talks to a sender via the API.
A Construct 2 game can run on the Chromecast (as the receiver) but you need another device (the sender) to control it. So if you implement an Android, iOS or Chrome desktop app that monitors keyboard, mouse, touch, voice etc, it can send custom messages to the C2 game based on what the user is doing.
In a modified version of Autorunner, I added a single event which detects when a message is received, checks if that messages is "jump" and if so simulates a jump.
c2astreceiver exposes the Google Cast API to Construct 2 but I don't currently provide any higher level functionality. Google also just announced new APIs at Google IO which I hope to take a look at this weekend.
The best place for a deeper understanding would be the Google developer site:
https ://developers.google.com/cast/docs/receiver_apps
You can see what Chromecast supports in terms of HTML5 by pointing it to:
https ://html5test.com
via the developer console.
In terms of Construct 2 and Chromecast compatibility... I can only say that I haven't run into an issue yet.
In regard to the sender app, maybe the sender app can be created using C2 (and another plugin) since C2 can run on Android but I haven't investigated that possibility yet.
I hope that answers your question.