I want to marry a girl America
No. Neither will automatically become citizen of the other country.
You do not need those records to get married in Mexico but you can call your state vital records office to check on that.
No, Mexican citizens can't get divorced in California if they were married in Mexico. One of the parties would need to be an American citizen.
No. Immigrants must fulfill certain requirements to become Mexican nationals.
"Mexican" is a nationality-- a Mexican is a citizen of Mexico.
Your Mexican passport.