Learn Hardcore Russian: Учитель разрешил мне задать вопрос. - The teacher allowed me to ask a question.

It’s the dative case. Verbs of permission like разрешить and позволить take the person who is being allowed in the dative: кому? — мне. The pattern is: разрешить/позволить/дать кому + инфинитив (allow/permit/give someone + to do something). Here: разрешил (allowed) + мне (to me) + задать (to ask) + вопрос (a question).

Why is the perfective infinitive задать used instead of задавать?

Задать (perfective) points to one complete action: “to ask (one question) once.”
Задавать (imperfective) is for repeated, habitual, or ongoing actions: “to be asking (questions), to ask (questions) in general.” So разрешил мне задать вопрос implies permission for one question now. If the teacher gave general permission to ask questions (e.g., during a Q&A), you’d say: разрешил нам задавать вопросы. Tip: Future forms you’ll use:
One-time: Я задам вопрос (I will ask a question).
Ongoing/general: Я буду задавать вопросы (I will be asking questions).

What’s the difference between задать вопрос and спросить?