So long as your paperwork with the embassy is in order, and the legal portions of the marriage are covered, it really makes little difference where you have the wedding ceremony, whether it be in Mexico, the US, Antarctica, etc.
No. Neither will automatically become citizen of the other country.
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.
yes
No. Immigrants must fulfill certain requirements to become Mexican nationals.
Most foreigners can obtain a Mexican passport if they move to Mexico with an immigrant visa and live there for four years (2 years if married to a Mexican citizen). Then one can apply for Mexican citizenship and passport.
"Mexican" is a nationality-- a Mexican is a citizen of Mexico.
No.
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 they do not the Mexican instead becomes an American citizen
You must move to Mexico with your spouse and live there for two years with an immigrant visa, then you can apply for Mexican citizenship. You can have both a Mexican and a US passport and be a dual citizen. If you were not married to a Mexican citizen, you would have to live there for four years.
Yes
She will still need to be sponsored by the spouse (who is an American Citizen) and apply for residency first. Once permanent residency is granted, she has to wait (I believe it's approximately two years) to apply for citizenship.