Learn Danish: Sagde bedstefaren hvem der hørte sirenen i morges? - Did the grandfather say who heard the siren this morning?

In Danish, when a question word like hvem (who) or hvad (what) is the subject of an indirect question, you must add the dummy subject der. In the clause hvem der hørte sirenen, "who" is doing the hearing, so it is the subject and der is required.

Would we still use der if the sentence was "Did the grandfather say who the boy heard?"

No, because in that case, "who" is the object of the indirect question, not the subject. The boy is the one doing the hearing. You only add der when the question word is the subject. It would simply be: Sagde bedstefaren hvem drengen hørte?

Why is "this morning" translated as i morges instead of denne morgen?

Danish uses fixed expressions with i plus an -s ending for parts of the day that have already happened. Since the grandfather spoke and the siren was heard in the past, i morges is the standard, natural way to say "this past morning."

Why does the sentence start with the verb Sagde?

Because the main sentence is a yes/no question ("Did the grandfather say...?"). In Danish, yes/no questions are formed by placing the verb before the subject, so we get Sagde bedstefaren instead of Bedstefaren sagde.