Koreans use Hanja and most people know a few common ones. But it's not widely used and fairly unnecessary to learn.
Edit: To clarify what I mean by "use". They do not write hanja or use it personally. It's largely limited to things like menus (used where we might use L/M/S to indicate size), parks/historical sites, and TV news tickers where space is limited. Hanja is kind of weird in that few people use it, but it's still used just enough for people to know the most common/simple characters..
Eh, they don't really use it. My wife is Korean and knows pretty much no hanja, though she knows some kanji since she took Japanese in high school (she moved to the states after high school, so no college experience in Korea). Hanja and kanji have some overlap, so I suppose she could recognize quite a few hanja given context.
They don't use it in every day reading, though it's used occasionally, typically with a hangul reading next to it (e.g. if it was on a menu or monument).
So, think of hanja as fitting in a Korean language history class instead of a modern Korean grammar class.
From what I understand, you're ideally meant to know about 100 as a minimum when you leave school in Korea. Most people seem to forget the less common ones after a while.
I think it would be great to know hanja as it would help me break down the meanings of hundreds of nouns more easily, but the workload is obviously immense.
Korean has no articles, at all, like russian. So there really is no difference other than the context where it fits. The course should technically accept alternate answers with the other article (except where it doesn't make sense of course) the course just reached the beta phase on Web though, so I'd give it a while, you can report too, that helps immensely.
Just because something can't happen in real life doesn't mean it can't happen in fiction. It's physically impossible (probably) so it (probably) can't happen in real life, but it's not logically impossible, so there's no reason why you couldn't find the exact sentence in fiction.
If you want to use 조용하다, the meaning would first of all change to "the bird is quiet and/therefore writes a letter" and you'd need to change the place of 편지 to be next to 써다. So that would make: 새가 조용해(서) 편지를 써요. But honestly, it is not a very natural way of saying things, because there is no causal relationship between being quiet and writing a letter. So short answer: no, that wouldn't make sense :)