You answered your own question: see illegally, as in unauthorized, omitting rules.
Always.
Yes.
Passports are now mandatory for US citizens leaving/returning to the US including both Canada and Mexico as of 2009. Usually you can get your passport taken care of at the post office and it may take up to 6 weeks if you do not pay extra for the expedited service.
You can if you have all the papers needed and that you are not staying there illegally.
yes
You only need a valid passport.
None other than the passport.
Yes. My cousin was about 9 months when she came from South America to U.S.A. She needed a passport then.
A passport is not needed for this trip.
It you are an American citizen, you don't need any document besides your passport, which is actually needed for your return into the United States. If you are not American and are reaching Mexico through the United States, you would need the American visa to enter Mexico; then you will have to request a Mexican visa that can be provided on the spot by the immigration agents.
In the past, you never needed a passport to visit Mexico if you lived in the U.S. But i have been reading in the news that the U.S. will be requiring everyone to have a passport to leave the country, no matter where you are going, wheather it be Mexico or any other foreign country. In regard to Mexico, I am not sure if you do need a passport to visit that country at this time, but will be required soon, if it is not already.
no you would need a passport and a birth certificate a letter is not needed