Actually if it’s simply a put the profits on the hedge stand independently of what they actually produce. They could lift the hedge at any moment or take the opposite bet to offset it regardless of their actual production. It seems to me it would be more advantageous to lift part of the hedge at a profit and then produce less since the price is way under the cost of production.

Depends on how elastic the oil price is.

Remember: producing oil costs Mexico. They can collect on the puts without producing oil.

If the oil prices drop below Mexico's production cost, whether it's useful for them to continue producing depends on the impact their have on world prices, and thus the payout for their puts.