USC sponsoring mother's GC

Question details

I am a US citizen and I am filling up forms for my mother's GC. She is currently in US on a visitor's Visa. There are a few issues
1. My last name and my Mom's last name is a bit different - thanks to my school which goofed it up. They are phonetically similar though. Will it cause any issues when applying?
2. I am trying to get a non-availability of Birth certificate from India. But I am running into brick walls. How important will it be in the GC application process?
3. After I file the GC papers will my Mom be able to stay beyond her current I-94 dates.

ANSWER

1. Names are phonetically spelled. So, I do not think this is any problem.
2. VERY important.
3. Yes.