Learn English – “the” with knowledge

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).

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.

I can see you've done a lot of research, and have come up with an ambiguous understanding which is a shame (English sucks). I hope I can help clear the air for you. :)

Your first instinct about there being no article needed in the first example was correct:

Without air and water, living things could not survive.

In fact, in most of the examples your sister found to support the being the correct particle, the could actually be removed entirely:

Birds and insects could not fly without air to support them. Without air, humans would not be able to fly either.

We can’t survive for more than a few minutes without air, so why isn’t air as much a part of us as our legs or arms?

To put into more simpler terms, if your body is dehydrated and you have just finished a tough gym session and have eaten a protein rich meal – without water the protein will never get to the muscles and therefore never get repaired.

In all of the examples above, "air" is being referred to more as a concept than as a tangible thing. We're not talking about a "specific air" that we could hold or touch or see. Also, it's not just a concept of one thing, but "some amount" of it. The sentences above aren't referring to one air, but rather an amount of air. However, like you said, it is uncountable, neither plural nor singular. I'll try replacing the word air with information. Again, we're not referring to any specific information, but rather the concept of some amount of information. I know, it's silly, and doesn't make sense...but it works, grammatically:

Birds and insects could not fly without information to support them. Without information, humans would not be able to fly either.

We can’t survive for more than a few minutes without information, so why isn't information as much a part of us as our legs or arms?

To put into more simpler terms, if your body is dehydrated and you have just finished a tough gym session and have eaten a protein rich meal – without information the protein will never get to the muscles and therefore never get repaired.

See? It works, even if it's nonsense.

In this example, things are a little different:

The sun, the moon, the sea, the sky, the Arctic Circle, the environment, the capital, the air, the ground, etc.

The reasoning is correct, that "the definite article is used in front of things generally regarded as unique." There is only one air being referred to here: the air on Earth.

So to sum up, think of the first example again. Is the sentence referring to an amount of non-specific air? Yes. So, we don't need to use the.