U.S. nationals, including U.S. dual nationals, must use a U.S. passport to enter and leave the United States. U.S. dual nationals may also be required by the country of their foreign nationality to use that country's passport to enter and leave that country.
The Mexican government only requires that your passport must be valid during the entirety of your trip. Please contact the airline(s) you will be flying on to ensure that they let you board the plane as the United States does have a six months validity rule for passports that may apply.
Answer: No, you cannot travel internationally without a valid passport. Photocopies of passports are not acceptable. If you need your passport to travel internationally and do not have time to wait for routine processing, then you can apply for expedited passport renewal. By mail, this process takes 2-3 weeks.
Yes, you will be denied entry abroad if your passport is about to expire. Additionally, if you need a visa, you won't receive one with a valid passport of less than six (sometimes three) months.
The Issue Date of your passport can be found on the data page of your passport book or on the front of your passport card. Some countries require that your passport be valid at least six months beyond the dates of your trip. Some airlines will not allow you to board if this requirement is not met.