Learn Turkish: Kadife paltoyu şimdi asıyorum. - I am hanging the velvet overcoat right now.

When a noun describes the material something is made of (like kadife for velvet, altın for gold, or tahta for wood), it acts just like a regular adjective. Because of this, it doesn't take the possessive compound suffix that you would normally see when combining two nouns.

Why is the accusative case (-yu) used on palto?

The accusative case marks a specific direct object. Because the speaker is hanging a specific overcoat ("the velvet overcoat"), you must add the accusative suffix to palto. Without it, kadife palto asıyorum would mean "I am hanging a velvet overcoat" (referring to any general velvet coat).

The verb is asmak (to hang). Is this the same verb used for hanging a picture on a wall?

Yes! Asmak is the standard transitive verb used for hanging clothes, pictures, or any other suspended objects. Just like in English, it is also the word used for hanging a person.

Could we use the aorist tense here and say asarım instead of asıyorum?

Not in this context. The word şimdi (now) tells us the action is happening right at this moment, which requires the present continuous tense (-ıyor). The aorist (asarım) would imply a general habit ("I hang") or a future willingness ("I'll hang it [later]").