Why no subject?
There's no need to use the pronoun, because the word form Хочешь already tells you the subject by the ending.
You can certainly do that in English as well: "Want some juice?". In Russian it is simply more common and less informal.
Should this not be. "Хочешь сокa?". As the "some" is implied and an amount is genitive.
You should certainly use genitive if "some" is indeed implied. On the other hand, if you are offering a glass of juice, you would use accusative: "Хочешь сок?"