"państwo" is rather "ladies and genlemen", for a mixed group (or sometimes, in writing, you don't know exactly who will read it so you write "szanowni państwo")
But apart from that, it's also a country. Well, "a state" could be a closer translation politically (and I don't mean a state like Texas, but a state like Venezuela or Burkina Faso), but in everyday language I think English just uses "country" and Polish doesn't care much about the technical differences between "państwo" and "kraj".
Could »Państwo« also be considered a translation to the German »Volk«, that is the populace? Or would Polish prefer a term like »Populacja«? Given your definition of »Państwo« as a translation to »Ladies & Gentlemen«, maybe it also refers to the people of a nation as well, thus creating a translation to the German terminology which is said to be hard to translate into other languages like English. (Which could bluntly use the term »Folk«, actually)
Thanks in advance!
Państwo as a noun is a political term that corresponds to German 'der Staat'.
'Das Volk' would be naród.
And populacja sounds much like '(Tier)population' to me. If you meant 'Bevölkerung', then it's ludność (ludność świata = Weltbevölkerung) or zaludnienie (gęstość zaludnienia = Bevölkerungsdichte).
Państwo as a pronoun is grammatically comparable to the old-fashioned German 'Herrschaften', as in:
Ich bitte die Herrschaften (=Sie) um Geduld. - Proszę państwa o cierpliwość.
The difference being that the Polish construction is not old-fashioned, as it's just as widely used as the German 'Sie'.
On a side note, when preceded by a modifier, the pronoun państwo becomes a noun, which means 'Ladies and Gentlemen':
Szanowni państwo! - Meine sehr verehrten Herrschaften! (in modern German '... geehrten Damen und Herren', of course).
Thanks a lot for the detailed response!
So, the Polish language seems to adapt those terminologies in a broader sense, as I would have thought that, as in German, terms like “Naród” (Nation) and “Rząd”... Oh! I see that I made a mistake. I thought that “Rząd” translated to Staat, but it is Government, apparently.
And yes, I did mean “Ludność”, but seemed to have falsely derived it from the English right into Polish. Funnily, German translators show few results, and a third of them always included the terminology “Gęstość Zaludnienia”, so I think that the former translation comes closest to what I mean to say.
On a side note, when preceded by a modifier, the pronoun państwo becomes a noun, which means 'Ladies and Gentlemen': [...]
So, it could be used as an alternative to Panie i Panowie? Under the precession of any modifier? Well, I will note that then. Thanks for this as well!
As for your last paragraph, I wouldn't say that verehrte is obsolete nowadays, it only sounds a bit more solemn; you might use this as an opening to a charity event or an exclusive soirée. While it is less common nowadays, I wouldn't say that it would be considered downright wrong; only a bit exaggerated in certain situations. As always in languages, context matters.
So, it could be used as an alternative to Panie i Panowie? Under the precession of any modifier? Well, I will note that then.
I now realised that I should have phrased that a bit better. Of course there is only a small set of modifiers that you can combine with państwo. Szanowni państwo is the most common one, drodzy państwo also works, although perhaps more commonly as an interjection surrounded by commas. Most modifiers would either make no sense, or change the meaning (nasze państwo = our state).
I wouldn't say that in this context Panie i Panowie is a real (common enough) alternative, though. The corpus says the following:
[base=szanowny] panie i panowie
And I would personally avoid the the first option, because I don't know whether I should use szanowni or szanowne. It depends around which words you mentally put the brackets, so that's rather tricky:
Szanowni (panie i panowie)
(Szanowne panie) i panowie
It's probably safer to split it:
Szanowne panie, szanowni panowie.
But that just gets really long and unnecessary. Just stick with państwo :D