Learn Polish: Czy Państwo chcą się napić dobrego wina? - Would you formal plural like to have a drink of good wine?

Not necessarily! While the na- ... się pattern usually means doing something to excess or satisfaction (like naczytać się, to read one's fill), napić się is slightly different. It has become the standard perfective verb for simply 'having a drink' of something.

Why is the wine in the Genitive case (dobrego wina) instead of the Accusative (dobre wino)?

The verb napić się requires the Genitive case. This is a leftover of the partitive genitive, implying you are drinking 'some' of the wine rather than the whole thing. As a rule of thumb, when a na- ... się verb takes an object, that object will almost always be in the Genitive.

Why is the verb chcą in the third-person plural ('they want') when the English translation says 'Would you like'?

The formal pronoun Państwo is grammatically a third-person plural noun. When you address a mixed group formally as Państwo, you must use the 'they' form of the verb, which is chcą.

Could the word się be placed immediately after napić?

Yes, się is somewhat flexible as long as it avoids the absolute beginning or end of a clause. Czy Państwo chcą napić się dobrego wina? is perfectly correct, but placing się between the two verbs, as done here, creates a very natural rhythm.