Learn Russian: Она правильно ответила на этот вопрос.

We use the perfective verb ответила because the focus is on the result and completion of the action. She gave the answer, and now the action is completely finished. If you used the imperfective отвечала, it would mean she "was answering" (focusing on the ongoing process) or "used to answer" (a repeated action).

English just says "answered this question." Why does Russian use the preposition на?

In Russian, when you state what thing you are answering (like a question, an exam, or a phone call), you must use на followed by the Accusative case. It helps to think of it as directing your answer onto the question.

Is этот вопрос in the Nominative case?

It looks like it, but it is actually in the Accusative case because it follows the preposition на. Because вопрос (question) is a non-living (inanimate) masculine noun, its Accusative form is identical to its dictionary form. The matching word for "this" (этот) stays exactly the same, too!

Why does the verb end in -а (ответила)?

As a quick refresher: in the past tense, Russian verbs don't change based on "I, you, or they" like they do in the present tense. Instead, they change to match the gender of the subject. Since Она (she) is feminine, the verb takes the feminine past tense ending -а. If a man had answered, it would be Он ответил.