It should be valid.
You can enter your country of citizenship as long as the passport is valid- even if it has 1 day validity remaining. However, as soon as a passport expires, you cannot board the aircraft using that passport.
there should be a date that says what day it expires
Yes, provided you have all border entry/exit stamps on your passport.
The first of June is the date you will need a passport, passport card, or enhanced driver's license to get back into the US from Mexico.
This makes no sense, but I think you mean if with an expired passport you can travel from Mexico to New Mexico or vice versa. In that case no, you need an ID or birth certificate if you are going to Mexico; if you are going to the US state of New Mexico from Mexico you need a passport AND a visa issued by the US government.
Certain countries require that there be a minimum time left on your passport before you can enter the country (eg 3 years to expiry) As these are generally countries ther require a visa to enter, the information should be given to you by the travel agent or particular embassy. But I don't think you can travel if your passport will expire before your return date unless you're an EU Citizen within an EU state because then there are ways to get a new one from home.
5 years for a child passport. 10 years for an adult passport
The US now requires a passport, passport card, or enhanced drivers' license to reenter the US from Mexico or Canada. Persons crossing into those countries will be checked for those documents to ensure they have the necessary documentation for return.
Yes, you are able to travel up until the day of expiration. >>>>>>>> This is not necessarily true. In most cases, the airline will not book your flight and/or let you travel if there's less than 6 months left on your passport.
If you are flying you need a passport. If you are driving over the border you can get a passport card, which is cheaper. If you are only going over the border for a day then I am not sure what you need.
From personal experience, you can go to the Mexican consulate in New York City. If you have all the required documents, you can receive the passport the same day, although you will have to wait several hours. The cost was about $110 for a passport that lasts five years. I think you can get a cheaper passport if you only want it for one year.