If a particular question could have multiple answers, you would use an answer. If a particular question has one answer, you would use the answer.
However, if you have multiple questions (as in your interview example), you have multiple answers (not necessarily per question - each may have exactly one answer or many answers). In this case you would use an answer, since there is more than one answer in the interview (but not necessarily more than one per question).
To address your examples:
At the outset of the interview students were told that if they did not know an answer they could say "pass" and move on to the next question.
There are multiple questions in an interview. It's possible they will each have only one correct answer; even in that case there are many answers throughout the interview.
It is always a good idea to go over the test to make sure that you answered every question. If you do not know the answer, guess. You may get the right answer or partial credit.
This one, I expect, is contributing to your confusion. Each question on the test has a single answer, but the test has many. The first sentence talks about the test as a whole, where it can be understood that the second and third sentence talk about a particular question (without making the transition very obvious, other than using "the answer" and "the right answer").
It can be a reasoning exercise in which the student has to figure out an answer on her or his own.
It's a reasonable expectation that a reasoning exercise may have multiple correct answers (or no correct answer at all), and that each student will likely come up with something different.
The teacher-librarian serves as a guide to help students figure out the answer on their own.
This likely refers to the general case of a student having a question. The student wants to find the answer to the question (or possibly an answer). For the general case of an unknown/unspecified question, the answer is usually used (at least I would, and that seems to be what I've seen), although an answer would also be correct.
A better statement of the definite article "rule" would be that you should use the X only when there is a reasonable expectation that your readers or hearers will understand what particular X is meant.
This includes situations when you have previously mentioned X; but it is not necessary to have mentioned a specific train, because few people hang around a railroad station with the plan of taking "a" train, any train that shows up. They're there to catch (or meet) a particular train. Even on a subway system where trains run on any given line every fifteen or twenty minutes, people are waiting for the next train.
Consequently, the hearer or reader may be expected to infer from context that what is meant is “the” train the speaker/writer is waiting for: the train to or from a specific place scheduled to arrive here at this specific time.
Best Answer
Context is the key issue here. “Make lunch” is the usual formulation, but “make a lunch” can work too in a particular context. He is talking in the video about making “a lunch” and packing “a lunch” in a lunchbox to take to work. He then has “a lunch” ready to eat when it is lunchtime at work. The indefinite article (“a”) here indicates that he has prepared one of the many possible lunches he could prepare for work - sandwiches, a tin of tuna, ramen, yoghurt, fruit, etc. It’s an example of a lunch.
He could easily have said all that he did say about lunch without ever using the indefinite article. It would have been just as correct and idiomatic to say “every day I make lunch to take to work” but in this context he chose “every day I make a lunch to take to work”.
It works just the same with breakfast. It’s usual to say things like, “Do you want me to make breakfast?” but in some contexts the indefinite article a can be used. For example, “He is a real pig. He makes her get up before him every day just to cook him a breakfast.”