"King of the Hill"

I think a simple random selection isn't fair. It would be better if players could choose randomly, but within a single round, a player shouldn't be selected twice.

This is closer to the truth. According to the current rules, the losing player gets a break for the next 3 matches; perhaps increasing this number to 4-5 would be optimal.

Also, the time for a single game should be limited, I think to 3 days, so that there are as many games as possible. And if you can't play faster, i.e., you can't defend the top of the mountain, then you should be removed from there.

I already wrote about this, that the game should be completed within a week or faster. If the King can't maintain the pace for whatever reason, then he falls into the pool, and a new pair is started. 3 days is still not enough; few people will be able to play at such a pace for a long time.