Why do we not use the instrumental case here?
because "nasz dyrektor" =our director is a subject of the sentence, and "gdzie" works like adverb.
Indeed, thank you!
"Gdzie jest nasz szef"? - why not?
"Szef" = "Boss", "director" doesn't have to be your boss in fact.
"Director" doesn't have to be a boss but he CAN BE a boss as well.
Is dyrektor considered as a masc personal noun?
Yes. Although it doesn't matter in this sentence.