If Kilo is tapped to station a permanent, its last ability will resolve before the station ability resolves.

When you proliferate, you can choose any permanent that has a counter, including ones controlled by opponents. You can choose any player who has a counter, including opponents. You can’t choose cards in any zone other than the battlefield, even if they have counters on them.

If a permanent or player has more than one kind of counter on them, and you choose for that permanent or player to get additional counters, that permanent or player must get one of each kind of counter they already have. You can’t have them get just one kind of counter they already have and not the others.

Players can respond to a spell or ability whose effect includes proliferating. Once that spell or ability starts to resolve, however, and its controller chooses which permanents and players will get new counters, it’s too late for anyone to respond.

You don’t have to choose every permanent or player that has a counter—only the ones you want to add counters to. Since “any number” includes zero, you don’t have to choose any permanents at all, and you don’t have to choose any players at all.