"The cat is drinking milk."
Translation:Kedi süt içiyor.
It's kedi for both "a cat" and "the cat", because it's the subject. Subjects can never have the accusative suffix (-yi,-i, etc.), so you can't show "the" on the subject. If it is the object, though, it is totally fine. For example if you had to say "I saw the cat" and "I saw a cat", you could use kediyi and kedi, respectively, to show the definiteness.
No. Kedi sütü içiyor is a correct sentence. But it's the translation of the cat is drinking THE milk (that you put in front of it.), so it's some specific milk we're talking about.
Now imagine you see a random stray cat that is drinking some milk. You don't know the backstory. In that case, you would say kedi süt içiyor. - the cat is drinking milk. Some milk.