A couple of clarifications:
Hopefully in the opposite order - upon marying the US citizen, one must be free to marry; if you’ve been married before, any previous marriages must have ended by divorce, annulment, or death of the previous spouse.
Here is some more procedural detail on the green card application procedures for the spouse of a U.S. citizen.
No, within the 3-month period before the end of the person’s second year of conditional permanent residence, based on marriage to a U.S. citizen, he/she must apply again to have the conditions on his/her permanent residence removed. He/she doesn’t need to still be married to the U.S. citizen; however, he/she will have to prove to the satisfaction of U.S. Citizenship and Immigration Services that the marriage was not for the purpose of evading immigration laws.
After the permanent resident has been a permanent resident based on marriage to a U.S. citizen, and has remained married to that U.S. citizen for at least 3 years, he/she will be eligible to apply for naturalization (citizenship). More details here. Naturalization isn’t by any means automatic; in addition to the waiting period as a U.S. citizen, the applicant has to pass an English exam, a civics exam, and a criminal background check, among other things. But you are correct that once a person has gained permanent residence through a valid marriage to a U.S. citizen, he/she will not automatically lose that permanent residence (or citizenship, if he/she has naturalized) simply by getting divorced.
Eva Luna, Immigration Paralegal and veteran of numerous marriage fraud hearings (as the court interpreter, not as the person engaging in marriage fraud!)