If the last time counter is removed from a permanent with vanishing and the sacrifice ability is countered, that permanent will remain on the battlefield indefinitely with no time counters on it. As long as the permanent doesn't have any time counters on it, neither of vanishing's two triggered abilities will trigger again.

If a permanent that's already on the battlefield without time counters on it becomes a copy of a permanent with vanishing, it will stay on the battlefield indefinitely (unless time counters are somehow put on it later). If a permanent with one or more time counters on it becomes a copy of a permanent with vanishing, it will vanish as normal. If a permanent enters the battlefield as a copy of a permanent with vanishing, it enters with the appropriate number of time counters and will vanish as normal.

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.

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.

If a player or permanent has more than one kind of counter on it, and you choose for it to get additional counters, it must get one of each kind of counter it already has. You can't have it get just one kind of counter it already has and not the others.

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.