"He has not given me anything."
Translation:On mi niczego nie dał.
Well, "żaden" (and its forms) is like "none", "not any", "not even one" etc. "no" is also often an appropriate translation.
"nic" is simply "nothing" (although sometimes the English translation is "anything").
"Żaden policjant mi niczego nie dał" = No policeman has given me anything.
Nie usually precedes verbs. If you choose to negate a pronoun instead, you create strong emphasis. Mi is a non-emphatic pronoun by definition, thus nie can't precede it. Furthermore, I'm pretty sure, when there's another pronoun like niczego involved, such a "negative pronoun" construction is impossible anyways.
Ничего не мне дал is definitely wrong in Russian, too.
I'm not sure whether this is in line with the linguistic consensus, but I claim that an accusative form of nic does in fact not exist, because it's a negated direct object by default. Nic and niczego are alternative genitive forms that are the same in terms of meaning and grammar, but there may be a difference when it comes to stylistics. As you can see in my other comment, I chose the shorter form nic, because it felt more natural in the middle of a sentence. I guess you could say that niczego feels a bit more emphatic.
Note that, if a preposition precedes this pronoun, you should always use the longer form (e.g.: bez niczego).
Thanks for your very informative explanation. I chose to use "nic" instead of "niczego" and was marked correct — not knowing why. The reason is now clear, both "nic" and "niczego" are genitive! It's also good to know that one should always use the longer form after a preposition! Thanks for the extra information!