Green card resident. short travel to Mexico. need re-entry permit?

So you don't unless more than 1 year:

A permanent resident alien returning to the United States from a visit abroad of less than a year may apply for readmission by presenting an Alien Registration Receipt Card ("green card") to the immigration authorities at a port of entry. The one-year time limitation does not apply to the spouse or child of a member of the Armed Forces of the United States, or of a civilian employee of the U.S. Government stationed abroad pursuant to official orders. In this case, the spouse or child must present the card mentioned above, not have relinquished residence, and be preceding or accompanying the member or employee, or be following to join the member or employee in the United States within four months of the return of the member or employee.

A permanent resident alien who intends to remain abroad for more than a year should, at least 30 days prior to the proposed date of departure, apply while in the United States to the U.S. Citizenship and Immigration Services (USCIS) in the Department of Homeland Security for Reentry Permit. The permit is valid for two years and may not be extended. If such a permit is obtained the alien may use this card to reenter the United States within the period of validity. Every alien applying for readmission must satisfy the immigration authorities that he or she is eligible in all respects for admission.

No, as long as your card is not yet expired and you will only be there temporaly.

While you will not need re-entry permit.

You may need additional documentation while travelling in Mexico. Check before you go too deep into the country.