It's up to the employer. That would be stupid on their part, for a green card is only the proof of one's lawful permanent residency, but not the residency itself. Can be renewed any time, unless in conditional status. Immediately file for renewal on Form I-90 and show filing receipt. If on conditional status in the aftermath of a marriage-based green card, then your I-751 receipt is the proof that your now expired green card has been extended by one year and that employment is authorized.
Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
They could but why would they? You are a LPR and have two forms of ID that satisfy I-9 requirements. If this is actually happening (as opposed to you worrying about it!) then have an attorney call them.
Due to the nature of this forum, I often do not have all the information required to provide legal advice. Accordingly, my responses on Avvo are intended as general and not legal advice.