Most of the time, Polish personal pronouns can be omitted, because in the presence of the conjugated verb, they do not add any new information. But they are really necessary to draw special attention to a specific person, or
to emphasize the contrast or the main difference between two elements:
CO lubisz? Lubisz mleko - You like milk (it's about milk)
CO lubię? Lubię wodę - I like water (it is about water)
Ty lubisz mleko - YOU like milk (it's about YOU, nobody else)
Ja lubię wodę - I like water (it's all about ME, ME, ME, ME, ME)
Ty lubisz mleko, ja lubię wodę - YOU like milk, I like water
Ty lubisz mleko, a ja lubię wodę - YOU like milk, and I like water
Ty lubisz mleko, ale ja lubię wodę - YOU like milk, but I like water
(The sentence is about the contrast/difference between YOU and ME)
Not exactly - it will be understood but it has slightly different meaning and sounds a bit weird: "lubisz mleko, lubię wodę". It is a bit like "you like milk - I like water - who cares" in everyday Polish. If you want to emphasize difference between my taste and your taste I would keep the pronouns: "Ty lubisz mleko a ja lubię wodę" - "you like milk and I like water (so get off my case - I am not driving to store to buy milk for you)".
These are different cases. "woda" is Nominative - the basic, dictionary form of the verb. It will mostly be used as the subject of the sentence.
"wodę" is Accusative, it's generally used for the direct object in the sentence. Numerous verbs simply 'need' Accusative, and "mieć" (to have) is one of them.
Technically there isn't one in this sentence (probably 'i' wasn't introduced yet), but I'm usually adding such an option because it just makes so much sense and is (more) natural. But it's not a suggested answer.
And actually, if there really was "and" in the English sentence, Polish should use "a" and not "i".
"a" is the contrastive "and". If "I like X and you (also) like X", there's no contrast, so you use "i". If "I like X and Y" it's also a very simple 'and', therefore 'i' is used. But here, "You like X, and I like Y". There is contrast. It has to be "a".
"lubicie" (you made a typo, although accidentally "lubcie" is also a valid form - the 2nd person plural imperative) is 2nd person plural. Like Castilian Spanish "vosotros", if you're familiar with it (if you learn Spanish only with Duolingo, then it's rather Latin American Spanish and they do not use it).
"lubisz" is 2nd person singular, like Spanish "tú".