Have you change your name, became a U.S. citizen or your immigration status changed?
No, none of those.
I just have some doubts, like what documents would I need to prove my identity.
Well, you would be asked for your U.S. immigration documents, like the I-551 Permanent Resident Card.
And for F-1 students?
F-1 students must show Social Security their I-20 Certificate of Eligibility for Nonimmigrant Student Status
what other documents would Social Security ask?
The I-94 Arrival / Departure Record in an unexpired foreign passport is another accepted document.
What if DHS doesn't authorize me for work after a name change?
in that case, we can change your name in our records but we can't issue a corrected card.
Can I work even if I'm assigned a non-for-work social security number?