Is there a way to assign assignments to just one (or a subset) of students?
Also, along the same lines, I have a couple students who did not join the classroom as soon as they should have, but who are in now. Their activity is showing that they have completed the required work, but it is not registered as "done" because they technically weren't assigned the assignment. Is there a way to remedy this?
The Duolingo for Schools system only lets you assign assignments to the whole class. A student can be a member of more than one classroom, though, so you could assign subsets of your class to other classrooms, just to give them those assignments.
XP assignments - no, it counts the XP done while a member of your classroom. You can choose whether or not to take any notice of the Duolingo for Schools screen, though - since you can see that they've done the work, you can mark their homework as done, regardless of what the DforS screen says.