Learn Icelandic: Konan greiðir sér og velur fötin. - The woman combs her hair and chooses her clothes.

The verb greiða (to comb) requires its object to be in the dative case. The reflexive pronoun for the third person (himself, herself, themselves) is sig in the accusative case, but it changes to sér in the dative case.

The English translation says 'combs her hair', but I do not see the word for 'hair' here. How does that work?

The phrase greiða sér literally translates to 'combs herself'. In Icelandic, this is the standard and complete way to say someone is combing their hair. You do not need to add the actual word for hair!

Why does the sentence use fötin (the clothes) instead of saying 'her clothes'?

When talking about personal items closely attached to a person, like clothes or body parts, Icelandic usually just uses the definite form of the noun instead of a possessive word. Because it is the woman doing the choosing, fötin is naturally understood as 'her clothes' in this context.

Why don't we need to add the word for 'she' before velur (chooses)?

Just like in English, if the same person is doing two actions connected by og (and), you do not need to repeat the subject. Konan (the woman) acts as the subject for both greiðir and velur.