Thanks, forgot to use vocative. Speaking of that, my sister's boyfriend (Łukasz) is a native polish speaker, and when I tried to get his attention by calling "Łukaszu" he was confused and said no one would ever get his attention like that in poland. Is the vocative dying out when calling people by name?
Yes, I believe it is. It sounds quite natural with diminutives of names, but with the basic forms of names it's rather weird. Although that's subjective, of course.
On the other hand, you can call someone in a semi-formal way (panie Łukaszu = "Mr. Łukasz"), and it's perfectly fine here.
No no no. Gotować - to cook (generally), but also more specifically: to boil. ugotować - 'to cook succesfully', finish cooking the meal. Gotowany - sometimes an adjective describing the way something was made - boiled, and not fried, for example (ryż gotowany/ryż smażony). Also I guess "sth being cooked/boiled right now", but it's hard to imagine using that, frankly. Ugotowany - cooked (ready). But that's probably also not used that often, I'd rather simply use gotowy" = "ready".
Oh, also one more thing: Czy jesteś już ugotowany, not 'jeszcze'. It can be 'jeszcze surowy' = still raw. I understand you wanted the English sentence to be "Are you cooked yet?" I almost always think of 'yet' as 'not yet', and this is exactly 'jeszcze' (jeszcze nie). So this happens in negations. But in questions, yet translates to Polish 'już', as it means more of an 'already'.
I guess that you could argue so because of Russian готовить, which means both "to cook" and "to prepare". And Polish has in fact the form "gotować się" which is an old-fashioned and rarely used way to say "prepare". But the basic verbs for "cooking" and "preparing" are, respectably, gotować/ugotować and przygotowywać/przygotować.