Is this genetive singular? If so, why not the accusative, kobietę? Thanks
Yes it is singular, genitive.
Because you need genitive after szukać.
(My interpretation - because NIE MA JEJ , and we need to JEJ SZUKAĆ,) - as you probably know negation usually needs genitive
It's all right to but lady instead of women or not?
Well, "lady" is not a basic word, while "woman" and "kobieta" are totally basic...
Also, "lady" isn't even as popular as "pani" in Polish.
"Are you seeking that woman" comes up as incorrect. It is quite formal, but not archaic.