Learn French: Le suspect est coupable, mais il le nie. - The suspect is guilty, but he denies it.

It stands for the entire idea that the suspect is guilty. Instead of repeating mais il nie qu'il est coupable (but he denies that he is guilty), French uses the neuter pronoun le to represent that whole idea.

If the suspect were a woman, would we say elle la nie?

No, it stays le! Even if we were talking about la suspecte, this le is a neuter pronoun. It isn't replacing a feminine or masculine noun; it is replacing the abstract idea of being guilty. The neuter pronoun is always le.

Can I drop the le and just say mais il nie, like we often drop "it" in English?

You really shouldn't. In French, the verb nier (to deny) usually needs a direct object. Even when English speakers simply say "he denies" without "it," French requires the le to point back to the thing being denied.

Is the word coupable related to the English word "culpable"?

Yes! They share the same Latin root. While "culpable" is a somewhat formal or legal word in English, coupable is the standard, everyday French word for "guilty."