Edit: Thanks to @Edwin Ashworth in the comments below, it appears that the answer to Part One is No, this categorization of "plural noun treated as singular" is not valid and is basically just an imprecision on the part of the New Oxford American Dictionary.
As Wikipedia notes (bolding added),
Plural in form but singular in construction
Certain words which were originally plural in form have come to be used almost exclusively as singulars (usually uncountable); for example billiards, measles, news, mathematics, physics etc. Some of these words, such as news, are strongly and consistently felt as singular by fluent speakers. These words are usually marked in dictionaries with the phrase "plural in form but singular in construction" (or similar wording). Others, such as aesthetics, are less strongly or consistently felt as singular; for the latter type, the dictionary phrase "plural in form but singular or plural in construction" recognizes variable usage.
My impression from this is that a noun that is "plural in form" merely looks like (rather than unambiguously "is") a plural: possibly it was formed via the usual process by which plural forms are produced, and it may or may not have been used as plural (in terms of verb agreement) in the past.
Or we could define "plural" by either form or function (morphology or syntax):
We can look at verb agreement to say when a noun is used as plural, or
We can look at the form of a word, disregarding how it is used.
These often match, but when they don't — words like "mathematics" or "billiards", for which we always say "mathematics is…", never "mathematics are…" — the right thing may be to use alternative clearer terms, like "nouns taking plural agreement" ("nouns plural in construction") and "nouns plural in form", rather than to call them plural as NOAD does. And of course, often these occur with non-count usage, so there is no question of calling them singular or plural, which makes the category of "plural noun treated as singular" even more suspect.
[Original answer (to Part Two) below]
I still haven't seen any examples that aren't -ics words (besides 'news'), but let me post a tentative answer for the -ics case: I remembered something I had quoted in the inning/innings singular/plural question, and looked up Fowler. Under -ics, the book says:
-ics. 1. Among the names of sciences, arts, or branches of study, are a few words in -ic that rank as real English; the chief are logic, magic, music, & rhetoric; but the normal form is -ics, as in acoustics, classics, dynamics, ethics, mathematics, physics, politics, tactics. The substitution of -ic for -ics …
2. Grammatical number of -ics. This is not so simple a matter as it is sometimes thought. The natural tendency is to start with a fallacy: We say Mathematics is (& not are) a science; therefore mathematics is singular. But the number of is there is at least influenced, if not (whether legitimately or otherwise) determined, by that of a science. The testing should be done with sentences in which there is not a noun complement to confuse the issue:—Classics are, or is, now taking a back seat ; Conics is, or are, easier than I expected ; What are, or is, his mathematics like? ; Politics are, or is, most fascinating ; Your heroics are, or is, wasted on me ; Athletics are, or is, rampant in the big schools ; Tactics are, or is, subordinate to strategy. The rules that emerge are: (1) Singular for the name of a science strictly so used; Metaphysics, or Acoustics, deals with abstractions, or sound. (2) Plural for those same names more loosely used, e.g. for a manifestation of qualities; often recognizable by the presence of his, the, &c.: His mathematics are weak ; Such ethics are abominable ; The acoustics of the hall are faulty. (3) Plural for names denoting courses of action or the like : Heroics are out of place ; Hysterics leave me cold. (4) The presence of a singular noun complement often makes the verb singular: Mathematics, or even Athletics, is his strong point.
So that long discussion very well illustrates the circumstances in which -ics words are sometimes used as singular and sometimes plural (though I wouldn't say "His mathematics are weak", for some reason). Note that it seems to be discussing when a certain -ics word is singular and when it's plural, not saying that it's always plural and discussing when it's treated one way or the other.
Anyway, along with RegDwight's useful answer, my hypothesis for my original question is this: if a singular word X exists (e.g. mathematic), and the word Y is formed from X by the normal rules of English pluralisation, then word Y is called "plural", even if it has never been used in the plural.
Of course, all this still doesn't answer the latter question of what purpose this category serves etc.