It's good to drain down and then fully charge when you first get it to train the battery. Then you can charge it whenever you want after that. It's also good to drain down and fully charge once a month to keep the battery healthy.
If I let my iPhone drain down until it turns off, after a while it turns back on when it has a little bit of power. I haven't tried powering it while it was off since I like being able to get calls while it's charging.
It's OK for it to be on while charging, but it will not charge as fast, since it's on and using power. I haven't checked the difference in charging time for on vs. off.