Form I-131

I am a brand new citizen and I am filing papers to bring my parents from England to NY to live with my husband and our four children. As I understand it once they get their interview and medical stuff out of the way, and get their visas do they enter as Legal Permanent residences?

My mother is currently in the US. She entered based on the Immigration Visa on her passport and gave the sealed packet at the port of entry. We went to to the SSN office but were told that we need to wait for the SSN to come (via mail? - they had no clue). She wants to leave in the next few weeks. Questions:
1. Can she reenter without a physical GC in hand? She was told it takes 6-8 months to come at port of entry.
2. How do we get her SSN? Is it mailed automatically?
3. What else should I be aware of before she leaves the US?

An applicant whose Form I-131 is filed and currently pending at a USCIS Service Center may go to a local office to apply for an expedited Advance Parole in case of an emergency. Have there been any changes to the process in which the applicant presents a copy of the receipts for the I-131 (and the underlying applications), documentation proving the emergency, filing fee, and two photos?