"I write everything down that ..." is incorrect, but why? Isn't that an acceptable word order?
"I write down everything that he says" or "I write everything that he says down" is much more natural, but I'd say yours is probably acceptable too.
After looking it up, it seems they are two different verbs with the same root. писать, and its perfective form (which I confused this for initially), написать, are simply verbs for "to write." записывать is actually a different verb, with the same root, with a meaning closer to taking notes or recording something. Please correct me if I'm wrong, but this is my understanding.
"I write everything down.."...was not accepted for me either. I don't understand; it seems the same to me.
I think сказать is perfective and you can't have that in the present tense