I am afraid I can't speak to entry to Mexico. That would be something you might want to ask your airline. Because they'll be checking passports before allowing you on an international flight as they don't want to be stuck getting you back if your papers aren't in order.
I can speak to using it in the US, however. Because I used my passport as my ID at the San Jose airport on literally the last day before it was due to expire and with no problems. This was with the TSA check in guy you have to pass through before you endure the security lines.
That has been my understanding - the airline doesn't want to have to deal with your being refused entry in Mexico, which could theoretically happen if you were to arrive with less than 6 months before your passport expires because you might have plans to remain in the country for up to 6 months on a tourist permit. As you know, not everyone is subject to that limit when entering Mexico on a foreign passport, but airline counter personnel aren't going to be very interested in that.
This can't be an issue when using a passport from the country of which you are a citizen to enter that country. You never necessarily need to leave again, and if your passport expires while within your country of citizenship, that will never present any sort of problem in terms of your remaining there.
Totally up to the airline. Good luck.
Whether you're allowed to board your departing flight from your home country is up to the airline (because there is no actual legal need for a passport in order to leave your home country). If you're leaving from a country other than that from which your passport was issued, their immigration people will want a record that you've left (so you do legally need your passport when leaving). If you legally have multiple passports, there can be a lot of variations of requirements. The airline doesn't really care about complexities, and they're the ones that decide whether you can board your flight.
Whether you're allowed to enter Mexico once you arrive is totally up to the Mexican immigration officer. The immigration officers don't care that your airline flew you to Mexico, and they're the ones who decide whether you get past their desk to enter the country.
Those two decisions might not necessarily match. You are highly unlikely to change the decision of anyone by correcting their misunderstanding of the rules. I make sure my travel documents match the most restrictive policy I'm likely to encounter, even if I'm certain it doesn't really apply to me.