Not yet. Every so often you'll see someone who's born to a seemingly nonmagical family, but it's typically because they've a witch or wizard great-grandparent that nobody knew about. And most all things do, yes. There are a few plants or animals that are utterly unmagical, but they're the odd thing out.

And no, no, it's not at all. It's mostly forbidden, to be honest; the only things taught are-- well, childish things. A spell to tickle, or turn someone upside down, or what have you.