Fafsa & parents' SSN

I’m filling out my parents’ information on Fafsa and I couldn’t help but ask. My mom gave up on her green card about ten years ago and still has her SSN. I tried to put in down but both of my parents say that she has no connection to the US anymore so it’s okay to just write all zeros for that section as you do to those who don’t have a SSN.
I wanted to know if it was okay to put down all zeros for that part even though she has a SSN.
Also when applying for colleges, I put down her maiden name (because that’s how it is on her passport) but on her SSN, it’s her married last name. Would that be a problem if I have to put her SSN?

Oh the webs folks weave.

Is your mom living in the U.S? What about your dad?

Are they still married?