At least in AE, we'd say that they were taking a make-up test.

to take a test later due to illness

They are still taking last week's test.
evtl.

I've never heard make-up test in BE. (Which doesn't mean to say it's never used.)

I would say:
They're taking the test they missed last week (because they were ill).

Re #4:
I am familiar with sit a test (BE, more often than to sit for a test), but in more official or higher educational situations (university, entrance exams, etc.). If this is just secondary education, I would be more likely to say take a missed test.

#3: makeup test/examination is OK in AE.
(A student may be given the opportunity to make up work that was missed.)