1. No, when you're plugged in and the battery's charged, the computer is powered off of the plug.
2. It's not a very good idea to leave your notebook plugged in ALL the time, so you should calibrate it (run out its battery and charge it back up again) about every month or two. But it won't hurt for it to stay plugged in while you're at home, as long as it's not plugged in for months on end, so your battery's fine, you don't need to take it out. 🙂