american english - Usage of "homework," "schoolwork," and "assignment" in AmEng for schoolwork given to students to do at home

I've always used these terms in a different way that @James McLeod is describing them. Perhaps they are regional differences.

Assignment: One singular task or project; large tasks may be broken up into smaller assignments. I may get one or more assignment per class. Whatever it is, it is thought of as a singular 'unit of work' when submitted or completed.
Complex, multi-part, or really large assignments are generally referred to as projects.

Homework: The collection of all the assignments I have to do at home

Schoolwork: The collection of all the assignments I have to do for school, regardless of the location of where it is done. Can be a synonym for homework, but less frequently used as such.

(Further clarification after reading some comments): An assignment can also be done IN school. Assignment just means 'one task'. Which is why a lot of people do use homework assignment. They're specifying that this assignment is to be done at home (or at least, not during that class)

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Not sure about other parts of the country, but I have never heard of homework used to describe not-related-to-school work done on or in the home.

Yes, Housework means work done for the house/home. This typically means tasks like cooking, cleaning, laundry, yard work, and small house repairs (when done on one's own home, this term is not used if professionals are called in).

I have never known a native speaker to confuse housework and homework, but if someone learning AmE were to mix the terms up, I do believe they would still be understood by most people.