Why is it wrong to use "titkár"?
it's not. the test is incorrect.
It wants to make reference to a feminine secretary using "titkárnő", in my opinion.
that is probably right, and it still does not matter.
secretaries, nurses, flight attendants CAN be male
well, a stewardess means a female, but that's another story.
secretaries CAN be male. so translating them to "titkár" is correct.
What's the difference between 'Igen, a titkárnő bent van' and 'Igen, itt van a titkárnő'? Is the second sentence incorrect or maybe it emphasizes another information?
You 2nd sentence is not incorrect - it's just not the true translation. It means 'Yes, the secretary is here."
Would it be incorrect to use "az" in this case? If so, what is the difference?
Az is used when the next word starts with a vowel sound, "t" definitely isn't one.