"Do you remember we have a math test tomorrow?"
Without context, there's nothing in the Chinese to imply that it's our test, vs your test.
Certainly; However, if you pay close attention to the part of the sentence 明天有数学课时 (lit. "tomorrow has math test"), it refers to both the speaker and the listener that they will take the test tomorrow. This is the possible reason that "we" is implicitly emphasized in the translation.
The point of excluding characters is to shorten the amount of saying. In this course, sentences like those emphasize how a Chinese speaker would say in real life.
I don't know. Could the parents of kids taking the math test have an identical conversation? Or the school janitor? I don't buy that the Chinese is implying that anyone in particular is taking the test.