"Mexican" is a nationality-- a Mexican is a citizen of Mexico.
Your Mexican passport.
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.
no they do not the Mexican instead becomes an American citizen
Yes. You only loss your citizenship if found guilty of treason.
go to mexico and bribe the government
Yes, an American citizen may sponsor a same-sex Mexican spouse for a visa, permanent residency and, ultimately, citizenship.