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.
It is typical in American legal documents to define and name the parties and thereafter to refer to them by their names without articles.
Thus:
CompanyName LLC complies with laws during providing the services for a
client.
would most likely say:
Provider shall comply with all applicable law during the provision of services to Client.
The contract becomes concluded between a client and CompanyName LLC
right after completing the registration process by the client.
This agreement between Client and Provider shall come into effect when Client has completed the registration process, assenting to all terms of this agreement.
Best Answer
It makes sense that we'd find the less common a length of in patents because patents are describing a very generic case.
Let's assume I've invented something that allows me to polish guitar strings, and I market this product as Strolish (short for string polish).
When telling someone how to use the product, I would probably use the definite article:
However, when I'm applying for my patent, I would use patent jargon. I'm no patent expert, but I wouldn't be surprised to hear my patent lawyer advise me to use different language when applying, including making the language as general as possible:
It is remarkable how many Google hits for "a length of the surface" were found in patents:
That said, it's quite hard to dissect an actual usage, because most patents are rather technical and difficult for the layman to understand – like this excerpt from U.S. Patent 7276737: