Panowie, panie

Am I correct in thinking thst 'panowie' is the plural of 'pan'?
'Panie' is the plural of 'pani'?
And 'państwo' translates as 'family'?

yes, but "Państwo" can be also used as a formal address (to a mixed group)
Ladies and Gentlemen, we gathered here - Szanowni Państwo, zebraliśmy się tu ....
How do you like (you in English as in plural) - Jak Wam się podoba / Jak się Państwu podoba?

Kowalscy / Państwo Kowalscy - (the) Smiths / Mr. and Ms. Smith

of course państwo means also "country, nation", but it's "inna para kaloszy"

correct


not correct - depends on context, examples:

Panstwo Kowalscy - Mr and Mrs Kowalscy
Prosze Panstwa (jakis wystep) - Ladies and gentlemen (some show)
Panstwo - country

... and probably some others ...

family = rodzina

Has anyone ever run across the now obsolete Państwo Edwardostwo Kowalscy?
That seems the equiavalent of Mr and Mrs Edward Kowalski -- something that makes every feminist cringe.