Choosing whether to model a variable as a confounder (fixed effect) or a random effect

Question

Let's say we have 10 students, where each student have been taught by one of three possible teachers. Then let us record how many hours each student spends studying and their scores in a given exam. We are interested in studying the association between study hours and score.

Here are my questions:

1. In this context should we model the effect of the teacher as a random effect (given students are "clustered by" teachers) or instead should we model it as confounder/fixed effect?

2. Is the answer to my question hypothesis dependent? e.g. If we want to model the association between score and teacher we should model the teacher effect but if we are interested in the effect of time studied on the score we should model the teacher effect as a random effect.

3. Taking this further, should I model the students (student ID) as random effects?

4. Is there a general rule of thumb that allow us to navigate this issue in more general contexts?

Answer 1

1. Yes, formally speaking teacher should a random effect but with only three levels estimation will be extremely problematic (i.e. how much we would trust a standard deviation out of a sample with just 3 items).
2. Yes, it is hypothesis dependent. But based on the initial information, teacher assignment was not explicitly determined.
3. We can model students as random effect only if we have a student take multiple tests. It might be more relevant to assume a student-teacher interaction effect. There are cases that we use single item random effect to account for over-dispersion in some models but that's different.
4. No really rule of thumb. Just reading relevant literature and critically evaluating it.