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.

If Park Heights Maverick has a +1/+1 counter on it and is dealt lethal damage, it will die before its triggered ability resolves and you proliferate. You won't be able to add a +1/+1 counter to it in time to save 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.

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 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.