Translation:Write slowly; I do not understand anything.
I assume someone might say this to a professor who is quickly writing stuff on a whiteboard and erasing it right after.
No, in this case, you'd say, Не стирайте, пожалуйста, я не успел(а) записать.
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".
I suspect you're right. The Russian team probably has pot going "OK, let's throw this one in there, how long do you think it will take someone to comment on it? Closest one without going over wins."
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.
No, it doesn't have that meaning. In the case you described we would say, "Пиши аккуратно - я ничего не могу разобрать" or simply "Пиши разборчиво"
I think it's purposefully to keep you on your toes, but I could imagine someone saying this to a teacher or a tutor who is making them copy down lots of stuff.
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.
"Dear Grandma, I'm writing this letter to you slowly because I know you cannot read fast."
To me this sentence makes sence. Many years ago when we were children we "wrote" letters on each others back, and the other had to tell the correct word.
I think the sentence makes sense if someone writes too fast it makes their handwriting very rushed and messy
Maybe when they write fast the writing is ugly and you can't understand anything.
Why is ничего genitive? It seems like a direct object, so would normally be accusative. Is it because of the negation of не понимаю?
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.