Without asking the school to violate their privacy laws, I don't think there is a formal way to find this out. Possibly ask some friends, or hide behind a bush and watch for them lol. Seriously thats a tough one. Depends on what the reason for your inquiry may be too. Its possible the administrators would be willing to help if this student were a potential threat.
Start with the department of vital statistics in the birthplace of the child. Request Birth, marriage, and Death records (they could be now deceased, sorry) If the child was signed away for adoption, you may have little results.
That is really a hard question to answer, I'm not sure that there is a legal way to do that. I guess the obvious, just call the school and ask the secretary a question like "Is _____ in class today"? She may simply check and answer your reply or tell you that they don't know whom you're asking about, you then apologize for dialing the wrong number.
I think I would find a different approach without involving the school to find the answers that I was looking for, the school is a dangerous place for an "outsider" to be lurking for obvious reasons.