As far as I know it can be used by anyone claiming citizenship by parentage, not only for children. It is typically used by parents for their children, but nothing prevents an adult from seeking this proof.
About the name change I am afraid you would need to do it either before or after the N-600.
By the way, I think this case is one of many of people applying for naturalization (N-400) and being rejected because of already being a citizen, which leads people to apply for N-600 to be completely certain that USCIS agrees with that interpretation. I have voiced several times that I think USCIS should issue a certificate of citizenship at the time of rejecting the N-400 application, or if need be do a request for further evidence and then issue the certificate. People shouldn't have to be submitted to the conundrum of being denied naturalization and then having to prove citizenship through N-600 and lacking enough documentary proof for the N-600 application.
