What is the function of "o" in this sentence? Is it the same 3rd person singular pronoun?
No, it's not. In this sentence "o" means "that" indicating "that cheese-o peyniri". Just pointing out that the woman eats that cheese, not this cheese kind of thing. Does this help?
Ok, I think so. Does Turkish have a three-level demonstrative distinction like Japanese? Here, there, and over there.
Yes, burada-here, şurada-there and orada-over there.
Ok, so, bu-şu-o. Perfect, thanks!
So why is it not Kadin su peyniri yer
"Kadın şu peyniri yer." is one of the alternative translations and it should be accepted as a correct answer as we included it.
So if the translation is that cheese why they put the i at the end of peynir?
I think because it's the object and its definite so it gets the accusative. Hope in right :)