Does dla take the accusative or genitive?
Is working for is not accepted?
It's an accepted answer, it should have worked.
The right answer should be modified to include "This" as an appropriate translation for "Ta".
When do we use dla as opposed to using z ?
Well, generally "dla" means "for" (especially for someone) and "z" is "from", "out of" and "made of".
But different languages take different prepositions, so for example some situations that need "for" in English will need a different preposition in Polish.
is the director also female in this case?
It is possible, but a female one would rather be "dyrektorka", so "pracuje dla dyrektorki".