Learn Danish: Hvad er dit efternavn?

We use dit because efternavn is an et word (a neuter noun). The word for 'your' has to match the gender of the noun it describes, so din becomes dit.

If I want to answer this, do I use min or mit?

You would use mit and say Mit efternavn er... for the exact same reason. The word for 'my' must also match the neuter noun efternavn.

Is efternavn a compound word?

Yes, it is made up of efter (after) and navn (name). It literally means 'after-name', which makes it easy to remember as last name.

How is the word hvad pronounced here?

The d in hvad is completely silent, so it sounds just like 'hva'. Silent letters are very common in short Danish question words.