The English translation looks strange; normally the question is asked without you, "Does it hurt?" and 'you' is implied. "Are you in pain?" is another possible translation.
Worse. The English means something else. It's asking whether something is causing her harm. If your dog regularly bites you, then it is hurting you. It sounds like the speaker is asking about that meaning of 'hurt', not about pain.
Why is "Does that hurt you?" incorrect? Is it worse than "Does it hurt you?" because "זה" here can only refer to a part of the body? Or should "that" be an acceptable translation? Thanks for the clarification!
According to Reverso Context "כואב לך" means:
Does it hurt, hurt you, it hurts you, painful for you, pains you, are you hurting, you are in pain, ails you
Does it hurt, are you in pain, are you hurting = asking if you have pain.
Does it hurt you = is something (action/person/object, etc.) being done to you that causing pain. Does it hurt you when it rains... (because it's not really common to say: does it hurt you). It sounds wrong.
First is asking about "hurt", second is asking if "it" is causing the hurt to you.
The translation isn't a good match for what people are saying the Hebrew is supposed to mean.