"Whose are the shoes?"
Your version is grammatically accurate & more natural as spoken Turkish.
Kim? "who?" + -in (genitive)
Kimin? "whose?" (pronoun)
Ayakkabı + -lar (plural) + -ı (accusative) + (n) buffer + -ı (possessive)
Çünkü Türksün, Türkçe gramerin anlatmayı bilirsin.
^ & lingot.
I'm not sure "Whose are the shoes" is actually incorrect, it's just that we'd never say it that way. I mean, it's still understandable, and you might say it if, for example, you wanted your song lyric to have nice internal rhyme, but in everyday speech? Nope. We say "Whose shoes are these?" (Or "those" if the shoes are far enough away that you'd have to point to them.)
Zdig6, in the English sentence, "the shoes" is the subject. So it can not be marked with the accusative suffix. There is no transitive verb here with a direct definite object, no action, only the verb "to be". Ektoraskan has a good explanation, in his answer to Sam_Sarsik, above: "Kimin ayakkabıları? = Whose shoes? It is not a sentence. I add: "Kimin ayakkabıları giyersin"? = "Whose shoes do you wear"? The verb "wear" is transitive with a direct object, so "ayakkabıları"=the shoes".
I am repeatedly posting the correct answer -- I'm looking carefully each time, but the page won't register it properly. I can't proceed from the page.