Learn Icelandic: Af hverju barði strákurinn hundinn?

The preposition af means "from" or "of" and requires the dative case. Hverju is the dative form of the question word hvað (what). So, it literally translates to "from what".

What is the infinitive form of barði and why does the vowel change?

The infinitive is berja (to hit). Even though it is a weak verb, it undergoes a vowel shift in the past tense, changing the stem vowel from an 'e' to an 'a'. For the third-person singular subject (strákurinn), this becomes barði.

Both strákurinn and hundinn end in -inn. Does this mean they are in the same case?

No, and this is a classic trick with masculine nouns! Strákurinn (the boy) is the subject in the nominative case (strákur + -inn). Hundinn is the direct object receiving the action, so it is in the accusative case. The accusative base noun drops the '-ur' to become hund, but the attached accusative definite article is also -inn, resulting in hundinn.