"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 (except after prepositions), 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 after prepositions, which require the genitive case, you should always use the longer form (e.g.: bez niczego, do niczego).
And after prepositions, which require the accusative case, the shorter form is mandatory (e.g.: na nic, o nic).
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!
"dać" takes a direct object (the object given, here: nothing) in Accusative, and an indirect object (the receiver, here: me) in Dative. The negation changes Accusative to Genitive, but Dative stays.
So that's about cases. As for the order, if you wanted to go with "He gave me NOTHING", that would probably be "On nie dał mi niczego". Feels natural enough, emphasis on "nothing" sounds probable. The sentence here isn't that strong, it's just "well, ya know, he ain't given me anything, simple as that", so it doesn't put "niczego" at the end. "mi" definitely can't be at the end (that would be "He has not given ME anything, he gave a lot of things to her!", plus even if you wanted to say that, you'd need the emphasized pronoun "mnie"), so the author of this sentence put it at the second place in the sentence, but "On niczego mi nie dał" seems to mean the same and feels natural, if not better.