Learn English – Regarded / Regarded as

@user178049 is correct in that "[distinguish] the good from the bad" is an expression. Though you could fit other verbs in there besides just "distinguish." For example, you can take the good and take the bad, and then you've got the theme song for an American sitcom from the '80s.

The point is really that taking out the articles changes the meaning a bit (or at least makes the meaning a bit more ambiguous).

One must learn to distinguish good from bad

is a perfectly acceptable English sentence. But that's not super hard for most people. Little kids generally understand the difference between good and bad, or right and wrong. The trick is applying the theory to the real world.

In the real world, you've got gray areas. You've also got things or people or ideas that might seem good or bad on the surface but are actually the opposite.

That's why we add in the articles. "The good" and "the bad" refer to the specific things in the world that are good and not bad, or vice versa – whatever those things may be. (So they're "specific," but only sort of.)

In answer to your other question about "distinguish," I would say it is okay here. That is just a simple infinitive. You could say "how to distinguish," though; that's okay too.

To me,

One must learn how to distinguish good from bad

implies maybe a slightly different emphasis than

One must learn to distinguish the good from the bad

But I'd say they both get at approximately the same idea.

1. The second pigeon flew…

This is the simple past.
We also learn a previous pigeon had performed the same action earlier.

2. …just as the first pigeon had flown.

This is the past perfect tense.
The original sentence could remain as it is, but a good writer will probably sense that writing the term pigeon, or repeating the same verb fly twice is redundant. In order to overcome this, a pronoun is needed to substitute “pigeon”.

2. …just as the first one had flown.

Now, I quite like this version but it is not included in the multiple choice. The closest equivalent is C) …one had flown away. The adverb away, in this context, means “further from a place, thing or person”, and “fly away” is a very common collocation. So, the OP could choose C).

2. C) …just as the first one had flown away

If the writer wanted to use a pronoun, and avoid repeating the same verb, the auxiliary verb, do, is used.

3. B) The second pigeon flew just as the first one had done

Here, the adverb away is not mentioned at all, if had been added to the first clause, then the second clause would fit perfectly,e.g. “The second pigeon flew away just as the first one had done.”

There is the construction do + so, where different forms of do so substitutes the verb, and its complement. e.g. a) The second pigeon returned to its coop just as the first one had done so. b) They asked me to revise the essay and I did so (= I revised the essay.) c) Dangerous currents. Anyone who swims here does so at their own peril. In the OP's example there is no complement in The first pigeon flew and therefore "so" is not required.

For more information about Do as a substitute verb, visit the Cambridge Dictionary Grammar website.

D) is incorrect because a noun or pronoun is missing:

2. D)…just as the first had done

The first what? It might be a white dove, or a duck for all we know.

So all this boils down to personal preference, and style. There is nothing grammatically incorrect with clauses B) or C), either one, in my opinion, is appropriate.