Small clarification, kyokushin rules dont much care what techniques you use or how they look. Be it a thai, tkd or a kyokushin kick, no matter how sloppy it is done it is perfectly ok as long as it gives result. Face punches and elbows to the head are forbidden though, but head kicks and knees to the head is ok. As are lowkicks. Unfortunaly, it is no longer allowed to grab the head while kneeing it. It used to be, but the japs was at a serious disadvantage due to their small size and changed it.

If you are knocked down (or seriously stunned) but get up immediately the other is given half a point or half a win. two half point and it is over. If you are knocked down and stays down the other gets a full point or win.
If noone is knocked down or both are downed once, the judges can give the bout to the dominant fighter.
The "dominant" bit is why some kyokushin fights ends up as slugfests without either giving a inch. Backing up may be taken by the judges that the other was being dominant.

If noone was dominant, scores are reseted and a extention round is given. And then another one. In tournaments the prelims can have a max of 3 rounds (and they usualy do, unless theres been a knockdown) and quarter fianals and up may have 5 rounds.

And about the ichigeki event. As Sandy said. A bunch of kyokushin fighters faced a bunch of k-1 fighters of different backgrounds (and admitedly-skill).
Off the 7 fights on the main card 4 was fought with k1 rules where everyone used standard k1 gear (trunks), and 3 was fought with modified kyokushin rules (to allow for the ring) and required wearing gi´s.
It was the same in the minor fight card. about half was k1 rules, the rest modified kyokushin knockdown rules.