1) After someone counters they lose their turn. They may release into an ability, as I have seen it happen before, but they may -NOT- try to break a Genjutsu, unless otherwise stated. So to answer the question: when it says, "LOSE THEIR TURN" it specifically means they lose their turn. But this case is special because stun rules can affect this counter-ordeal.
2) From what I've been told, per round. But in instances it's per hit. That needs to be ruled, and I will speak with Haku about that later.