Agree with those above. Assignment is a good catch-all.
Homework can definitely be used as a general term, but to me, it often implies "due the next day" or at least a short term deadline. Say, a math assignment to practice what they learned in class and to be turned in at the beginning of class the next day.
A paper could be long or short, but is mainly a writing assignment. A "term paper" or "research paper" implies a long paper with lots of research and/or analysis that is worked on over a period of days/weeks/months. It doesn't refer to a test/exam in the US, other than what is known as a "blue book exam", where a high school or college student would write out lengthy answers to essay questions. (Incidentally, a "quiz" tends to be a shorter test, while an "exam" is typically a big, more comprehensive test.)
A project is typically of longer duration, but includes more than just writing. It could include making a poster, building a model, giving an oral presentation, etc.
My husband and I reference these terms a lot in conversation when it comes to my son's homework. I'm a great daily homework parent - knowing what assignments are due on which day, reminding the kid to get off the computer until he has his assignments done, etc. But I *hate* dealing with projects, especially if it's all "come up with something creative". My husband, on the other hand, would never think to ask if there was daily homework to be done (let alone check to make sure it's finished and actually in the backpack), but he loves projects and has a ball helping with all the creative stuff. And I'm happy to give him that responsibility, LOL.
