Once Obstinate Gargoyle has been blocked, causing it to gain flying by modifying it won't cause it to stop being blocked.

If a card with persist is removed from the graveyard after it dies but before the triggered ability resolves, it won't be returned to the battlefield.

If a creature has +1/+1 counters and -1/-1 counters on it, state-based actions remove the same number of each so that it has only one kind of those counters on it. A creature's persist ability can bring it back again if its -1/-1 counters are removed this way.

If a creature with persist that has +1/+1 counters on it receives enough -1/-1 counters to cause it to be destroyed by lethal damage or put into its owner's graveyard for having 0 or less toughness, persist won't trigger and the card won't return to the battlefield. That's because persist checks the creature as it last existed on the battlefield, and it still had -1/-1 counters on it at that point.

An Aura controlled by another player does not cause a creature you control to be modified.

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.

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