We know she's a woman, but we don't know if she's a lady ;-)
You would only normally translate nainen to lady in the address "hyvät naiset ja herrat" = "(dear) ladies and gentlemen". (Although I've also heard the Finnish one as "hyvät naiset ja miehet", but that was admittedly from a representative of a political party on the left...)
Yes, "on" in Finnish means "is". There are no articles in Finnish. The familiar things tend to be in the beginning of the sentence, and the new information goes towards the end. Colloquially "yks(i)" and "se" are sometimes used similarly to the indefinite and definite articles though.