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 of counters on it. Lesser Masticore'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.

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

You must discard exactly one card to cast Lesser Masticore; you can't cast it without discarding a card, and you can't discard additional cards. If Lesser Masticore enters the battlefield without being cast (perhaps because of its persist ability) you don't discard a card.

Once persist returns the creature, it's considered a new object with no relation to the object that it was. Auras that were attached to it won't return to the battlefield. Equipment that was attached to it will remain unattached. Any counters that were on it won't be put on the new creature.