You don't have to choose every permanent or player that has a counter, only the ones you want to add another counter 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.

While proliferating, if you choose a permanent or player with multiple kinds of counters, the permanent or player gets another counter of each kind, not just one kind. This change from the original proliferate rules was introduced in a previous set.

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 cast this spell by paying its entwine cost, you will be able to proliferate the counters that you placed with the first mode.

If a permanent has +1/+1 counters and -1/-1 counters on it, they're removed in pairs as a state-based action so that the permanent has only one of those kinds of counters (or none of either kind of counters) on it.

To proliferate, you can choose any permanent that has a counter, including ones controlled by opponents, and 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.