Do make sure that it is a legitimate ceremony and marriage certificate. But yes they can, as long as that American citizen is not a felon, is able to adequately support that spouse financially (affidavit of support) and when they come back to the U.S. gets down to the business of filing their spouse's paperwork.
no they do not the Mexican instead becomes an American citizen
The Mexican must apply for Resident Status and after 3 years of being a Resident Alien they can apply for Citizenship.
Not automatically. They have to apply for a green card and then apply for Naturalization.
No.
no, he needs to get a 3 year stay
They will become an American citizen
== == In Short: no.
Not automatically.
sure you can return back to the US
Yes. Effective June 26, 2013, an American citizen may sponsor a foreign, same-sex spouse for permanent resident status.
Once a person from another nation marries a Canadian citizen, he or she becomes a permanent resident. The process takes approximately 50 days for the paperwork.
US citizenship cannot be lost by her becoming a Mexican citizen. Mexico would control whether she becomes a Mexican citizen. US has no involvement in the process.