Mexico
Having a passport and/or Visa is a legal way to enter the U.S. from Mexico.
No
Your birth certificate will not help you. You can enter the US from Mexico with an "enhanced" US drivers license, but only if you are traveling by sea or land, not by air.
As long as you have a US visa with you, you can also enter Mexico.
To gain Arizona and New Mexico from Mexico
Yes. Also a US visa.
No. Having a visa for the US doesn't grant you the possibility to enter Mexico; you need the corresponding visa from Mexico.
Canadian, American and European citizens can enter Mexico without additional requirements other than their passports.
Canadian, American and European citizens can enter Mexico without additional requirements other than their passports.
Your passport; if you are not from the US, Canada or the European Union, you also need a visa.Arizona, New Mexico, and Texas border Mexico.
You don't need a visa to enter Mexico if you are an American, Canadian or European citizen. If you, however, are from other country, even when entering from either territory you need to have a visa issued by the Mexican government.