Learn Icelandic: Lánar hún músina sína? - Does she lend her own mouse?

Because she is lending her own mouse. In Icelandic, when the subject of the sentence (hún) owns the object being acted upon, you must use the reflexive pronoun sinn (here in the form sína). If she were lending another woman's mouse, you would use the invariable pronoun hennar.

Why is it músina sína instead of just mús sína?

When using a possessive word like sinn, the noun it attaches to usually needs the definite article (the 'the' ending) attached to it. So the noun mús first becomes músina (the mouse), making the phrase literally 'the mouse her own'.

Why is the possessive word exactly sína here?

The reflexive possessive pronoun sinn must match the noun it describes in gender, number, and case. Mús is a feminine noun. It is in the accusative case here because it is the direct object receiving the action of the verb (it is the thing being lent). The feminine, singular, accusative form of sinn is sína.

Where is the word for 'does' in this question?

Icelandic does not use helper verbs like 'do' or 'does' to make questions. Instead, it simply swaps the subject and the verb. By putting the verb lánar (lends) before the subject hún (she), it automatically becomes the question 'Does she lend?'.