Translation:Write slowly; I do not understand anything.
This is such a weird sentence. "Speak slowly", I could understand. But when you're writing it's not like the words vanish moments after they're written.
Unless it's a joke, like when you comment on something and say "Here, I'll type this response slowly because I know you're an idiot".
My theory is that it actually means write carefully, I don't understand anything, meaning that I can't read your handwriting so you should go slower. This would be said by a prison guard taking a false confession in the gulag who is watching as the prisoner writes it out.
I could see this if it were "write clearly" or "write carefully", if the writer is jotting things down in a hasty, illegible scrawl. But just "write slowly" with no reason why except "I don't understand" makes no sense. It's writing, you can read it at your own speed.
You are right. The particle ни, which became a prefix in negative pronouns like ничего, is only used to enforce negation, so, historically, it is similar to “any [of that stuff]”, hence the genitive. Cf.: Я кое-что об этом знаю (I know something about it) and Я ничего об этом не знаю (I don’t know anything about it). Я не знаю ни слова по-тамильски (I don’t know a word in Tamil). Whenever a direct object starts with ни, the following noun is always put in the genitive case and the pronoun ничто becomes ничего.