For instance in chapter 2 when Rikudou first explains the situation to new arrivals, he beheads one of the fighters with one swing of a sword - not damage, but kills instantly. Kyle notes the same thing in chapter 3 when trying to protect Nazuna.
Yes, the former is the reason he decided to talk to him. The latter is the condition on which he managed to get Beago to work with him. He didn't know about Beago's apprehensions before seeking him out. Remember: Kyle has to have a partner to compete in the trial.