Yes; especially if you are taking a flight to get to Monterrey.
If you live in Mexico then no.
Yes
No. New Mexico is part of the United States, so you don't need a passport. :)
Yes, you always need a passport when you travel outside your country of origin.
No. Both are in the United States. You do not need a passport to travel from state to state.
Yes.
Yes
Yes, you do.
Yes, they do.
Yes you do
To fly into Mexico you HAVE to have a passport. As of right now the only way you can get into Mexico without a passport is by land travel, but that will change as of Summer 2009. As of Summer 2009 you will need a passport for any land travel into Mexico and Canada.
No. American citizens don't require a visa to enter Mexico. All you need is your passport.