Learn Portuguese (Portugal): Depois da aula eu estudo.

The phrase for "after" is depois de. But in Portuguese, you usually need to include "the" before a noun. Since aula (class) is a feminine word, "the" is a. When de and a bump into each other, they merge to form da. So, depois da aula translates literally to "after of the class".

If this sentence is the answer, what "when" question is it answering?

This lesson covers the question word quando (when). If someone wanted to ask you "When do you study?", they would ask Quando estudas? (informal) or Quando estuda? (formal). This sentence is the perfect reply!

Do I need to include the word eu (I)?

No, you can just say Depois da aula estudo. The ending of the verb estudo already tells us that "I" am the one studying. Native speakers often include the eu just to make the sentence flow better or to add a little emphasis, but it is completely optional.

Can I use aula to talk about a whole course I'm taking?

Not usually. The word aula specifically means a single class session or lesson (like a one-hour math class). If you want to talk about an entire course or program of study, you would use the word curso.