Tromell, Seymour's Butler

While proliferating multiple times, players can't respond between proliferating the first time and proliferating the second time, and so on.

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.

The value of X is determined only once, as Tromell's last ability resolves. To determine that value, look at the nontoken creatures you control and count each one that entered this turn. It doesn't matter if they were creatures when they entered or who controlled them when they entered.

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.

If you proliferate multiple times, you don't have to choose the same set of players and/or permanents to get additional counters each time.

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.