Learn Icelandic: Tekur þú sængina mína?

Because it has to match the noun it belongs to! The word for duvet (sæng) is feminine and it is the direct object being taken, which puts it in the accusative case. Therefore, the possessive pronoun must take its feminine accusative form, which is mína.

sængina means "the duvet". Why do we say "the duvet my" instead of just "duvet my"?

That is a common pattern in Icelandic! When you place a possessive word like mína right after the noun, the noun must have the definite article attached to it. So you literally say "the-duvet my" to mean "my duvet".

Why does the sentence start with Tekur (take) rather than a question word like "Are" or "Do"?

To ask a simple yes/no question in Icelandic, you just swap the verb and the subject. Instead of saying Þú tekur (You take), you say Tekur þú (Take you...?). Icelandic does not use helping words like "are" or "do" to form questions.

What makes sængina accusative in the first place?

The verb taka (to take). The thing being taken is the direct object of the sentence, and in Icelandic, most direct objects are forced into the accusative case. This case change applies to both the noun (sængina) and its attached possessive pronoun (mína).