How do I apply for a green card? Instructions?

Your mother should be able to renew her greencard on line here https://www.uscis.gov/green-card/after-green-card-granted/renew-green-card Just tell her to ask for a sticker to be put on the old card which makes it valid for another year when she goes in for her biometrics. That will allow her to use the old card for both jobs and travel. It can take up to 9 months for a new card at the moment because there are so many applicants.

Yes, it is perfectly possible that her old employer will not ask her to show her greencard again. Her permanent resident status does not expire (that is the meaning of the word permanent). The biometrics expire unless you have a card issued before 1985. So once the employer has seen a valid greencard, that may be sufficient for him. She can also show him the USCIS confirmation of renewal.

I think that what your mother meant was that her employer may assume that her previous GC is still valid, so they may not ask for it again. But they should if they follow the law.

Without a valid green card your mother is illegal

http://www.uscis.gov/greencard

Your mother had to apply to renew her green card at least 6 months before it expired. By law, she is required to carry her green card on her person at all times. She is subject to penalties, fines, for failure to do so. And no, she cannot work without presenting the proper documentation & filing I-9 form.

Permanent residents of the United States eighteen years of age or older must carry their actual valid green card at all times. Failing to do so is a violation of the Immigration and Nationality Act, carrying the possibility of a fine up to $100 and/or imprisonment for up to 30 days for each

https://www.uscis.gov/i-90