The solution offered is not natural English. You could say "Do I know you gentlemen?"
"Panów" is dopełniacz/genitive of "panowie", so you're right: this sentence is plural masculine. "Państwo" (plural mixed) would be "państwa" instead.
Yes indeed. For mixed group it would be "Czy ja Państwa znam?" and for group of females: "Czy ja Panie znam?"
They're both questions.. I think if you wanted to ask the Duolingo question informally (ie. 'Znam cię?), you'd have to phrase it 'Czy ja cię znam?'. But I could be wrong :)
The position of the words may vary, and 'czy' and 'ja' are optional. Znam cię? is completely natural.
Znam cię would be correct, but it is not a formal way. In here, the question is a formal way of asking "Do I know you (plural masucline/mixed group)?"
It's plural, to more than one 'sir'. So "Do I know you, sirs?" would work (I think that's rare), as well as "Do I know you, gentlemen?".