Learn Portuguese (Portugal): Quem é o professor?

No. Unlike many words in Portuguese, the question word quem (who) doesn't have masculine, feminine, singular, or plural forms. It always stays exactly the same.

How does the word order here compare to a normal statement?

A standard statement usually follows a Subject-Verb-Object order. But when you ask a question using a question word like quem, that word goes at the front, followed directly by the verb, and then the subject: Quem (Who) + é (is) + o professor (the teacher).

Why do we use é here instead of está?

Both mean "is", but we use é (from the verb ser) because someone's identity or profession is considered a permanent trait. You use forms of estar (like está) for temporary states, such as locations or moods.

How would I ask this if the teacher is a woman?

You would ask: Quem é a professora? The question word quem and the verb é stay exactly the same, but you change the masculine article o to the feminine a, and add an -a to the end of professor to make it feminine.