Just curious - what do you usually use Chromecast for? I got it specifically to stream Youtube on my TV, and found that there was usually a > 1 second lag between the image on the TV and the sound on the TV. This has given me a pretty negative view of Chromecast - basically that it's just good for streaming visual-only things like slideshows etc. Have you found this not to be the case?
I think you're missing the Chromecast button on the YouTube page. You shouldn't have to mirror the tab to fling YouTube to the TV.
Mostly I use it for Netflix, Google Play Movies & TV, and Plex.
Also if you have an Android or iPhone the native application on either device will have the Chromecast icon to fling the video. It's the same icon in Chrome (and maybe other browsers? I'm not sure) to fling from your computer.
Definitely not. Since Chromecast is sending audio and video over the same connection (HDMI), this is likely a problem with your TV or the like. Do you have a complicated Home Theater setup?