Dan asks: "How long are the rounds, and how many are there?

since there is no break by the ref after contact, do they fight
uninterrupted until someone is down, and then a count begins?

also are there fights ever stopped without either a ko or time expiring? in other words are tkos ever awarded in Kyokushin?"

The first round is 3 minutes. if no one wins it, a extra round is ordered, then another, then if noone still has a win they check how much the fighters weighs, and howmany boards they broke in that segment. if there is still no winner a final extention is ordered. So there is a maximum of 4 rounds.
Each extention is 3 or 2 minutes (slight differences depending on organisation).

As for fighting, it is technicaly uninterupted until someone goes down, although since there are no ropes it does get interupted when someone is forced out of the fightingarea (if you do it enough times you get a penalty for fleeing, and the other guy gets half a point -remember a full point is a win), also he fight is interrupted if the fighters get stuck in a clinch.

I dont think I have ever seen a fight ended in the middle of a round exept for when one has been downed once for more than 5 secs or twice for less than 5 secs. But a down is not neccecarily a KO. Any time a fighter is rendered incapable to fight by a technique it is concidered a down.

A special case is when a fighter sweeps or throws the opponent. if he follows it up with a good focused (but controlled -no full contact) finishing technique (usualy restricted to punches, but some organisations allows kicks too) to the fallen fighter he gets a half-point as if the guy had been knocked down for less than 5 sec. No follow-up =no score, and it gets no more attention than a slip.