Answers
As legally present and residing in Greece, you are entitled to be accompanied by your family members. To prove the relationship you have to submit among others the following:
a recent certificate of marital status of foreign authorities officially validated and translated, from which the kinship can be deduced; for category C3.2 in the case of concluding a cohabitation agreement, which must have been concluded in Greece, a registration document of concluding a cohabitation agreement of a Greek public authority is submitted; for category A3 in the case of concluding a cohabitation agreement if it has been concluded in Greece, a registration deed of concluding a cohabitation agreement of a Greek public authority is submitted. Under a cohabitation agreement concluded in Greece, within her legal presence in Greece (3 months as US citizen), your girlfriend will be entitled for her PR according to family reunification provisions.