Learn Russian: Почему директор отказал нам?

The verb отказать (to refuse) requires the Dative case for the person being refused. This is why we use the Dative нам instead of the Accusative нас. You can think of the logic as "giving a refusal to us."

Is отказал a perfective or imperfective verb?

It is perfective. We use the perfective отказать here because the question is about a single, completed action with a clear result: the director made the decision and the refusal happened. Its imperfective pair is отказывать.

If the director were female, how would this sentence change?

The noun директор is grammatically masculine, but it is commonly used as a job title for both men and women. However, the past tense verb must match the person's actual gender. For a female director, you would add a feminine ending to the verb: Почему директор отказала нам?

Does the word order matter here? Could I say Почему нам директор отказал?

Yes, you can say that! Russian word order is very flexible. Placing нам at the end of the sentence puts a slight emphasis on who was refused. Moving нам earlier is also completely natural and shifts the focus back to the action of refusing.