I think I would translate it like "don't you want some water?" Because that is what feels natural to me. But I do hear some older people here in Oklahoma say "You not want some water?" But I can say that it is more rare for someone to say this than "You want water?" instead of "Do you want some water?"
I also thought "Want some chicken?" would be okay, and got it wrong. I feel like the 'do you' is normally implied, especially in informal situations. There is no 'Ты' or 'Тебя' here where we could infer the 'do you' from. The literal translation feels like 'Want (some) chicken?'.
Usually we don't say курятина. Купить курицу, готовить курицу, есть курицу (buy, cook, eat chicken).
My advice: if you see wings/chicken legs/other specific parts of the chicken, you should say курица. If you see only the meat (without specific parts that could let you know it's chicken), well, in this case you can say курятина.
— Это говядина или свинина (Is it beef or pork?)
— Нет, это курятина (No, it's chicken meat.)
You're right, "Would you like some chicken?" is the polite way of saying this and would rather be translated into a way more polite Russian, like this "Bы хотите курицу?" or "Хотите курицу?". "Хочешь курицу?" is the same as "Ты хочешь курицу?" which is a little less polite.