"My little cat died."
Translation:Küçük kedim öldü.
That cat is not the direct object here :) It is the subject of the sentence. (In fact, the verb "to die" can never take a direct object. "I am dying my friend" makes not sense....the form that takes a direct object would be the semantically similar verb "to kill.")
You only use the accusative when a direct object and when specific.