Well, a bit. You control the playback with your phone, but the Chromecast gets the video from YouTube or Hulu etc. directly. Maybe a $50 Roku stick is a more direct comparison.
In many cases, yes. But it also mirrors tabs and it's how I watch Amazon Prime video on my TV since they don't offer Chromecast support (since they sell a competing device).
So instead, I just cast the tab fullscreen from my laptop or desktop. That sort of thing doesn't really work well enough for full motion video on a mobile device but a "regular" computer handles it about 99% as well as a native Chromecast app.
You need access to a service on your phone/tablet/computer to push it to the Chromecast.