Why not 젓가락을 주세요 since it's the noun that the verb refers to.
Because these participles can be omitted when doing so won't cause confusion. The sentence in the exercise is a typical one as "juseyo" indicates the chopsticks are the object.
Yes, but that's hardly consistent. Duolingo often marks an answer wrong if the object marker is omitted.
Agree that it shouldn't be marked as wrong (typo).
I wrote 젖가락 좀 주세요. Isn' t that acceptable?
Yes that's probably more natural.
No, that would mean,"give me little chopsticks"
저에게 젓가락을 주세요 why this is not accepted