Learn Icelandic: Hundurinn meiðir köttinn.

Because the dog is doing the action to someone else (the cat). The active base verb meiða means to hurt someone or something. If we used the -st form (meiðast), it would mean to get hurt or to hurt oneself. For example, Hundurinn meiðist means 'The dog gets hurt'.

Why is the word for the cat spelled köttinn instead of kötturinn?

The cat is receiving the action (getting hurt), so the word must be in the accusative case (the object form). The base word köttur becomes kött in the accusative. Then we add the masculine accusative 'the' ending (-inn), resulting in köttinn.

Both hundurinn and köttinn end in -inn. Does -inn do the exact same job for both?

They both mean 'the', but they are in different grammatical cases! For the subject hundur, -inn is the nominative (subject) ending. For the object kött, -inn is the accusative (object) ending. It just happens that the masculine 'the' ending is -inn in both of these cases.

If I change the word order to Köttinn meiðir hundurinn, does it mean 'The cat hurts the dog'?