Yes. Whoever draws the fourth side of a box claims it and must move again, and keeps going for as long as each line closes a box.
Yes. Completing a box earns a bonus move, and the bonus repeats. If your next line also closes a box, you take that box and go again. Your turn ends only when you draw a line that does not complete a box.
Why this matters
It is why handing your opponent a box can be a mistake: they take it, then draw again. A long run of boxes can be taken in a single turn, as explained under what a chain is.
A single line can also finish two boxes at once. See whether one line can close two boxes. For how to use this, read the strategy guide.