Learn Russian: Он задал вопрос учителю.

In Russian, you cannot use спросить (to ask) together with the word вопрос (a question). Instead, you use the set phrase задать вопрос, which literally means "to assign a question" or "to pose a question". You could also just say Он спросил учителя (He asked the teacher) without using the word for "question" at all.

How would I change this sentence to say "He asked him a question" using the pronouns from this lesson?

You would replace the noun учителю with the dative pronoun ему. The sentence becomes Он задал ему вопрос. The person receiving the question is the indirect object, so they always take the Dative case, whether it is a noun or a pronoun like мне, тебе, or нам.

Why does the word учитель (teacher) change to учителю?

Because the teacher is the person receiving the question, the word is in the Dative case (the indirect object). For masculine nouns that end in a soft sign (-ь), like учитель, you drop the soft sign and add -ю to make it Dative.

If вопрос (question) is the thing being asked, why didn't its ending change?

Вопрос is the direct object, so it is in the Accusative case. However, for masculine nouns that are inanimate (non-living things), the Accusative form looks exactly the same as the dictionary form. Therefore, вопрос stays exactly as it is!