Learn English – “We have a game today” vs. “We have the game today”

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 am surprised that you, a native speaker of English, find this use of How is/was X? odd or novel. I have heard it all my life:

How's your sandwich?
How was school today?
How's the new Stones album?
How's college?

And a little Google-booking carries it back to at least the early 19th century, in a very colloquial novel of manners:

How was the Opera that night? We will give the account of it which appeared in the paper of the following day. —Charles White, Almack’s: A Novel, 1827, p. 146

“Pray, how is the tea tonight?” —ibid., p 179

It seems to me this is a standard form for a question inviting a description or assessment. Extending it to new video games is obviously a recent development, but it's hardly novel or localised.