You mean the Gadsden Purchase? The original objective of James Gadsden was to acquire land for a transcontinental railroad.
What not of them ? I think you need to be a little more direct in the question.
Yes
You could if you're in a country near Mexico, if you aren't, then you need a transport.
If you are from the US, Canada or the European Union, you don't need a visa. If you enter Mexico by land, all you need is a birth certificate or ID. Otherwise, you will need a passport.
A passport if flying there or a driver's license if going by land.
Yes. Also a US visa.
With a population of 116 million in 2013, I think you need to be a little more specific (i.e: there are more than 50,000 "martin dominguez" in the phone book).
more fresh water yes
Who is 'they'? You need to be more specific.
As a foreigner? Yes. There is some additional paperwork you need to complete, but you can.
The U.S. has a land border with Canada on the north and Mexico on the south. Need a link to a map to see it? You got it.
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.