Will be of great service to a lady, who's gonna marry some Slavic man)))
"Znów" and "znowu" are perfectly interchangeable. "Jeszcze raz" is literally 'once again', 'one more time', so use it when this literal 'one time' would really make sense. Here it wouldn't really work, but in "Przeczytałem tę książkę jeszcze raz" = "I've read this book one more time", it does.
"Znów" and "nów" are related and are derived from the word new in archaic Polish. "Znów", "znowu", znowóż" are very similar to Russian "снова" (pron. snova), meaning once over, from the beginning="od nowa" and have their root in the old slavic "nowj", which also is the root for the new moon "nów": https://obcyjezykpolski.pl/now-nowiu-nie-nowia/
It should be. It's not common but if somebody was really annoyed that you were drinking again (probably the wife) for perhaps the third night in a row she could say AGAIN you're drinking! Which sounds more like a telling off, where as 'You are drinking again' sounds more like a statement disguised as a question.
Hmm... Perhaps this should be considered yet another meaning of "jeszcze". "jeszcze" doesn't mean "one more time" on its own, but "Jeszcze raz" definitely does.
Or when your team scores a goal the fans yell "JESZCZE JEDEN! JESZCZE JEDEN!" hoping that their players will score another one quickly.