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.
Perhaps you mean uncountable nouns.
Usually, you should find a handful of uncountable nouns in any student's grammar books. As for the complete list, I'm afraid that there is no such thing. (If there is such a list somewhere, I would like to know too.)
In my opinion, memorizing them might not the most effective way. I usually consult dictionaries when I am in doubt, and you can do that too. However, you will have to choose well, for only some dictionaries will tell you clearly which noun is countable and which is uncountable. (Also note that several of them can be either countable and uncountable, depending on your specific use.)
For such an online dictionary, I recommend Macmillan Dictionary. For example, you can look up the word corruption at: http://www.macmillandictionary.com/dictionary/british/corruption. It will tell you exactly that corruption is uncountable.
Also note that, the is a definite article; a and an are indefinite articles. A singular countable noun must have a, an, or the in front of it. (Well, that's the basic idea, because there are exceptions such as my, your, that, etc.) For plural countable nouns and uncountable nouns, you can use either the zero-article or the definite article the.
The usage of definite article vs. indefinite article might be too complicated for me to cover all of them here. My suggestion is reading these posts: https://ell.stackexchange.com/questions/tagged/indefinite-article.
NOTE: There are quite a handful of questions in ELL being tagged with articles, definite-article, indefinite-article, and zero-article.
Best Answer
This answer addresses the two questions at the bottom of the body of your question, but only tentatively addresses the question in your title.
The grammatical construct where the article is missing appears to be called the zero article. There is also a book written about the zero article.
One might have thought that zero articles are used when the noun is inherently unique, since there is no need then for an article to distinguish any one of many from a specific one of many. However, English is inconsistent here. One example that has come up in a similar discussion is The Eiffel Tower. Another example (from the book, IIRC) is The Baltic Sea. In both cases, the noun is unique but the definite article is part of its name.
One suggestion is that we always use the zero article with names. In examples such as The Eiffel Tower and The Baltic Sea, the word The is considered to be part of the name rather than a separate article. Street names are similar. For example, "Where is Main Street?" has no article but "Where is The Strand?" does because The is part of the name The Strand. Note that this doesn't cover all cases since "I can design an Eiffel Tower" and "I can pave a Strand" are arguably acceptable. In many cases, though, this 'rule' holds.
In the case of your question, Room 401 is the name of the location, but the location is not so grand as to warrant having "The" as part of its name. The definite article is therefore not used before Room 401 in the sentence in your question's title.