A:
I'm not sure about the internet speed requirements, but I can explain a little about how the device works. It doesn't get plugged into your computer, it gets plugged into your television's HDMI in. The chromecast doesn't store anything. You use a separate device (Android, PC, iPhone/iPad) and 'cast' to the device. What this does, essentially, is send the URL for the video you are watching to the device and from there it picks up the URL and streams it.
Some of the apps/websites that work with the chromecast are Netflix, Hulu+, YouTube and HBOGO. There are quite a few more and with just a little googlin' around and reading forums and such, you can find even more. I personally use XBMCcast and Show Box (neither available in the market, but easily found and sideloaded onto an android device via a google search for XBMCcast apk or Show Box apk) coupled with Localcast or Bublble UPNP (both found in the Google Play Market) to watch just about anything I want to watch, including the latest episodes of popular TV shows.