Those things have specifically been tested and worked on in beta, I don't think a retail customer would have the kinds of problems you're worried about.

The battery has a thermal sensor so you shouldn't be able to cook it even if you try. And the player will shut itself down before the battery goes totally dead. But even if you do manage to make the battery totally dead (there's a way to do it if you really try, but I won't tell you how ) we specifically tested to make sure it would still charge from that state.