I already understand and so ask NOT about the definition, below which I want to burrow. I heed the Etymological Fallacy. 1. Which ODO definition corresponds? What does of mean here?
to be of = Possess intrinsically; give rise to
2. How does the juxtaposition of be + of effect/imply/induce this meaning?
In its oldest senses, of meant something like “away from”, which of has lost but is preserved in off (“come off the porch”, “what a rip-off!”). It comes from the same Indo-European root as the Latin word ab, which roughly means from or “the source of”.
The general meaning of indicating where something came from, or coming away from something, is still roughly preserved in most of the modern senses of of: indicating the source or origin of something (cuisine of Russia), a reference point for something else (north of Boston), the characteristic period of time when something happens or exists (the Rite of Spring, flowers of summer), something that is removed (cured of the hiccoughs, deprived of liberty), the cause of something (proud of his accomplishments, sick of eating rice), the result of something (a doer of good deeds, the wages of sin, the product of effort), and many more, including the senses probably most relevant to your question: material (made of mahogany), belonging or possession (territories of the United Kingdom), and properties or qualities (the speed of the wind, hair of red). These latter senses indicate something about the object which explains behavior. If you imagine that the quality that the object is “possessed of” exists deep within the object, and its behavior is more superficial or external, you might have found the right way to see various other senses as extensions of the primary one.
So:
to be of foul temperament = to hold, deep within oneself, a foul temperament, which could give rise to foul behavior; similarly, to be of good cheer.
to be of age = to have reached a certain age, with the emphasis on the inner qualities that normally attend having reached that age.
to be of Russian descent = to have an ancestral origin in Russia.
to be of use = to be useful; to be among the useful, “of” the category of useful objects or people.
to be of the opinion that … = to hold that opinion; to be among the kind of person who holds that opinion; to have many other, consequential opinions that flow from the deeper opinion explicitly stated; to have come to that opinion from many experiences; for you, that opinion came out of much previous experience and thought.
Time is of the essence = An important aspect of the situation is that speedy action will be rewarded and dawdling will miss important opportunities. Of roughly indicates the source or inner nature; essence explicitly means what is most central or fundamental; so, to be “of the essence” means to be most definitely and unmistakably embedded in what is most central or fundamental.
I don’t think there is any one definition or sense of of that will uniquely explain these. Each can appeal to a different sense of of, to multiple senses simultaneously, or to the vague, general gist of it.
The word be just converts the prepositional phrase into a full predicate including a verb so you can use it where you need a verb:
A man of foul temperament was hired to manage my department today. So, as of today, my boss is of foul temperament.
Ben Kovitz has provided plenty of good examples, to which I might add the clear difference between "You are very kind" and "That's very kind of you."
The former is simply a statement of your general temperament, whereas the latter means you have chosen (from—of—many different possible ways to treat someone) to be kind to me instead of ignoring my plight or treating me with contempt. The second statement is considered 'polite' because I have recognized your choice and am grateful for it.
Another obvious example would be "You're stupid!" and "That was stupid of you!"
In the first sentence there is no hope for you; in the second you can return to your normal intelligent self in the future.
