Is it not possible to say "this is my boss"? Perhaps if you were introducing her to someone? I put this and got it marked wrong.
I agree. That is a very common phrase used to introduce one's boss without mentioning gender.
This is my boss=Αυτό είναι το αφεντικό μου.
Το αφεντικό is masculine, but this boss is a woman. Why shouldn't this be a feminine form?
"Το αφεντικό", as you wrote has the "το" article, so it is neuter. But, there are some professions that stay the same in masculine and feminine forms. For example ο/η γιατρός, ο/η δικαστής etc. ;)
Why is chief wrong? I am not a native English speaker, is there a difference between chief and boss?
It's better to use "αρχηγός" (which also means "captain") for "chief".
I hear "afentiko", not "afediko". Is there any rule, how can I undestand which sounds to use? when ντ is "d" and when "nt"?
There's no "nt" sound in Greek, it's either "nd" or "d" and always "d" at the beginning of a word.