But you can't just mirror a display to it, can you?

Yes you can - the UX isn't all that obvious though: Open Chrome, click on Cast extension, click on small arrow and select "Cast screen". It'll mirror your desktop to CC.

While you can do this, it means your device needs to be on (drawing power, running the radio, not on standby) for the whole stream, so the "normal", preferred model is for the content server to stream directly to the chromecast, and your phone is just the remote control for that process.

Exactly how this works is up to whoever writes the cast apps that you're using. APIs are over here: https://developers.google.com/cast/docs/developers

No. You can mirror a Chrome tab; that's the best you can do. And it doesn't work as smoothly as a purpose-built app.

You can mirror a whole display. It's an option in the Chromecast plugin for Chrome. It's labelled "experimental" but it actually works really well.