If Costume Closet has left the battlefield or has no +1/+1 counters on it by the time its activated ability resolves, you won't put a +1/+1 counter on the target creature. If the creature becomes an illegal target or can't have a +1/+1 counter put onto it for some other reason, you won't remove a +1/+1 counter from Costume Closet.

An Aura controlled by an opponent does not cause a creature you control to be modified.

A creature that is equipped is considered modified no matter who controls the Equipment that's attached to it.

Once Costume Closet runs out of +1/+1 counters, it remains on the battlefield. However, activating its second ability won't do anything.

The +1/+1 counters on Costume Closet won't affect it unless it somehow becomes a creature.

A creature with a counter on it is considered modified no matter what kind of counter it is or which player put it on that creature.