have you change your name or became a U.S. citizen?
Yes, I changed my name.
then you have to inform Social Security so they can update your records and issue a new card.
can I use my non-work Social Security number for work?
no, you can't. If you do we'll inform DHS.
I want to apply for work authorization. I'm legal.
You can apply for a work authorization either in the Department of Homeland Security DHS , U.S. or in  Citizenship and Immigration Services USCIS and request a Social Security card from the Social Security Administration.
and for F-1 students what documents are needed?
You'll need to show us the  I-20 Certificate of Eligibility for Nonimmigrant Student Status.
What I'm legally able to do if I still don't have permission to work.