I think you mean that English grammar does not allow double negatives, and that Russian, Italian, Spanish, and quite a few other languages not only allow them, but require them as part of the grammar.
When colloquial English uses double-negatives, they don't cancel each other out, either, but also intensify the negation. Example: "I don't have no idea what you're talking about!"
(Correct English: "I don't have any idea what you're talking about.")
Personally, I'd like to see certain double-negatives allowed in English, because they do indeed intensify the intent of the sentence.
Occasionally, I try to answer the question without looking at the clue boxes. I also thought it was "I don't read anything." When I saw the clues offered, I went with "I read nothing." I'm starting to see why some native Russian speakers I've met speak English the way they do.
I have a few thoughts about it:
1. The negation of the verb using не should be before the verb, so не читаю is something you can regard as standard.
2. Я as the subject comes before the verb, so Я не читаю is a word order you can rely on.
3. That leaves ничего, which needs to go either before or after the negated verb. Some people have said that words like ничего are pronouns, which would naturally place them before the verb, but I did a couple context searches and found examples where it was after the verb. There is probably a nuance of emphasis which I just don't understand yet that depends on the placement of ничего.
a very bad audio. I couldn't catch the last word at all, even I was playing it in turtle speed. And after I have been practicing the same many times before. For me it sounds like the speaker says: "Я ничего не четайм". Of course I have never heard about this word and it probably doesn't exist. But I could not recognize the right one at all. So DL, please improve some of your pronunciations!!! You don't need to bind the sounds together like natives would do. It doesn't help us rookies here.