What is the name for a woman who courts a man?

Answer 1

There is nothing unusual in a woman courting a man, although the convention is that it is the man who has to propose marriage. (And even then, convention can be defied.) If the woman in question is significantly older than the man she courts, then the current slang would call her a cougar. This term is more about the age difference than the courting, however.

When a man courts a woman he is a "suitor". When a woman courts a man she is a "suitress".