Learn Hardcore Russian: Учитель задал студентам вопрос о книге. - The teacher asked the students a question about the book.

задаёт is the imperfective present (“he is asking/poses habitually or right now”).
задавал is the imperfective past (“he was asking/posed repeatedly or over a period of time”).

What is the difference between задать вопрос and спросить?

Both can mean “to ask a question,” but they focus on different aspects:

задать вопрос (кому?) literally “to set/assign a question to someone.” It’s often used in formal or educational contexts (teachers, exams).
спросить (кого? о чём?) literally “to inquire of someone about something.” It’s more general and colloquial: you спросить друга о дороге (“ask a friend about the way”).

Why is студентам in the dative case while вопрос is in the accusative?

Russian marks the recipient of an action with the dative case and the thing being acted upon with the accusative:

студентам (dative plural) is the indirect object (“to the students”).
вопрос (accusative singular, identical to nominative here) is the direct object (“a question”).
So Учитель (nominative) is the subject, задал is the verb, студентам is “to whom,” and вопрос is “what.”

Why do we use the preposition о with книга and what case is книге in?

After verbs and phrases meaning “to speak/ask about something,” Russian uses о + prepositional case.
Here о книге is о plus книга in the prepositional singular (feminine), so книге means “about the book.”

Can we use про instead of о here? Are there any differences in tone or register?

Yes.

о книге is neutral and slightly more formal.
про книгу is colloquial and common in everyday speech.
Both mean “about the book,” and neither changes the core meaning in this context.

How flexible is the word order in this sentence? Could we say Учитель студентам задал вопрос о книге?

Russian word order is relatively flexible thanks to its case system. All these convey the same idea, with slight shifts in emphasis:

Учитель задал студентам вопрос о книге. (standard S–V–IO–DO–Prep)
Учитель студентам задал вопрос о книге. (focus on студентам)
Студентам учитель задал вопрос о книге. (focus on студентам or учитель)

What nuance does the prefix за- add to дать in задать вопрос?

The base verb дать means “to give” (imperfective). The prefix за- in задать makes it perfective and introduces the idea of “setting” or “assigning” something once. In задать вопрос, it emphasizes the act of “posing” or “putting” a question to someone as a complete event.

Could we remove о книге and still have a correct sentence? What would the difference be?

Yes. Учитель задал студентам вопрос.
This simply means “The teacher asked the students a question,” omitting what the question was about. The grammatical structure and meaning remain perfectly correct; you’re just leaving out the detail “about the book.”