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I am filing GC for my mother who is in US on visitor visa & her I-94 got extended for another 6 months. She never used her surname on her Indian passport? What should I mention in the last Name on I-130 & rest of the forms (I-485).Should I mention NA or not used or leave them blank?

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?