Is it permissible to send religious advice via e-mail to a doctor who is not religiously committed, without revealing the sender's identity as a girl?

The default rule concerning advising a teacher is that it is sought after, so long as it does not lead to discord or corruption; this is because it falls under enjoining good and forbidding evil. Allah, the Exalted, says: (You are the best nation produced for mankind. You enjoin what is right and forbid what is wrong and believe in Allah). The Prophet, peace and blessings be upon him, said: "Religion is an-nasiha (sincere advice)." If the advice leads to corruption, then it is not permissible, because avoiding prohibitions takes precedence over fulfilling commands, as the Prophet, peace and blessings be upon him, said: "If I forbid you from something, then avoid it, and if I command you with something, then do of it what you are able." It is not necessary to mention the name in the message; rather, not mentioning it might be preferable.