The book reports are often project-based, yes. However, the oral and written comprehension class is where a lot of the actual writing is done.
Re: notice of the unit assesment, No clue. I think this is one of those things that kids are told about and are supposed to write down, but I suspect that doesn't always happen. I know my own kid can tend to get a little loosey goosey with writing down what he is supposed to. Re: classwork, my kid's binder is CRAMMED with old wadded up classwork, so I think it must eventually come home.
You're probably already doing this, but I suggest combing through the binder nightly first thing, and really probing about any blank spaces on the daily agenda.