Contagion Clasp's first ability is mandatory. If you're the only player who controls a creature, you must target one of them.
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 on it.
To proliferate, 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 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.