"Io non ho una mucca, ho il suo cuoio, il suo latte, la sua carne."
I think so. In fact I think another accepted answer:
I don't have a cow, I have his leather, his milk, her meat
should be wrong. It's a bit weird for the graders because english typically uses gendered nouns for posessives (his/her) regarding people but not things (because things don't have genders in english). Animals are especially hard because many would refer to them as "it" in this context but pets usually would get genders.
You are correct. With animals one is to always use "its" for the possessive. Personal pronouns, possessive or otherwise, are not allowed even for pets; however in common slang today people often use personal pronouns of all kinds for their pets.
I think that people misapply personal pronouns to pets because they misapply personal characteristics to pets. This is a psychological projection of human characteristics to animals.
It's a grammatical exercise. So grammatically, you should continue to put "its." Additionally, even in English, you should continue to repeat the possessive in a list; without it a situation could arise out of the ambiguity that the last two items in the list are really appositives for the first item. While it doesn't make logical sense here to read it that way, it would make grammatical sense to read it that way. Here's an example of such an ambiguous case: "I really liked their performance, dancing and singing" vs. "I really liked their performance, their dancing and their singing." In the first example, I could have just really like their performance which included/consisted of dancing and singing. In the second, I really liked the three aspects: their performance, their dancing, and their singing.