The War of the Spark set doesn't include any ways for players to have counters, but some cards in other sets say that a player "gets" a counter of a certain kind. Notably, emblems aren't counters.
You can choose any permanent that has a counter, including ones controlled by opponents. You can't choose cards in any zone other than the battlefield, even if they have counters on them.
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.
Players can respond to the 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 a spell or ability puts counters on a player or permanent and then instructs you to proliferate, choosing that player or permanent will give that player or permanent another of those counters.
Putting loyalty counters on planeswalkers doesn't cause their loyalty abilities to be activated.
EDH / Commander: Restricted
Pauper: Ej legal
Premodern: Ej legal
93/94: Ej legal