Duo currently rejects "What are you gentlemen doing?" with good reason, but surprisingly proposes "What are you guys doing?".
In UK English, at least, 'guys' is highly colloquial and thus in conflict with the formal/polite 'państwo'.
Indeed I might, if I were an hotel commissionaire or a butler admonishing a boisterous group of tipsy young gentlemen – or if I were P.G. Wodehouse writing about the incident.
A valid objection – except that young mixed groups rarely trash the furniture, since young ladies, even if tipsy, exert a calming influence upon young gentlemen.
Well, "państwo" is a Formal You form used when you talk/refer to people of both genders at the same time. Other Formal You forms can somehow be translated into English (you, sir/you, madame/you, gentlemen/you, ladies), but we haven't found any natural translation of "państwo", apart from "you" itself, obviously. "you, people" doesn't exactly seem to be a polite form.
So in a way that is more polite than saying "those people/men/women"? Well, you can use the Formal You pronouns as nouns, but then they need some additional word, usually simply a determiner. And then you have "Co ci panowie robią?" (What are these gentlemen doing?) and similar.
As I answered in another discussion, "państwo" is problematic, as it doesn't have a good translation. You wrote "ladies and gentlemen", but it's very likely that it is said to just one lady and one gentleman, or two ladies but one gentleman, or one lady but two gentlemen...
Meaning aside, I don't recall ever hearing such a construction in English.