Learn Romanian: Femeia este profesoara. - The woman is the teacher.

You don't need to change the word order at all! Just say Femeia este profesoara? and make your voice go up at the end. In Romanian, changing your tone of voice is all it takes to turn a statement into a question.

If I wanted to ask who the teacher is, how would I use the word for 'who'?

You would ask Cine este profesoara? (Who is the teacher?). The word cine means 'who', and just like in English, it goes right at the beginning of the question.

Why do both femeia and profesoara end in the letter -a?

In Romanian, the word for 'the' is attached to the end of the noun instead of being a separate word before it. For feminine words, it is usually an -a. So femeie (woman) becomes femeia (the woman), and profesoară (teacher) becomes profesoara (the teacher).

Could I just say Femeia este profesoară instead?

Yes, but it changes the meaning slightly. Femeia este profesoară means 'The woman is a teacher', which describes her profession in general. Using profesoara means she is 'the teacher', pointing out a specific person you are already talking about.