How and when did impossible become a noun (as in 'To achieve the impossible')? Was it before or after impossible was an adjective? Why? Or were they formed from Latin simultaneously?
Impossible itself never become a noun. This is actually a common construct where "the + adj" is used as a noun.
EDIT: As RaceYouAnytime pointed out, impossible was used as a noun in the past. But to the example sentence provided, "impossible" is not a separate noun, but "the impossible" is a construct where an adjective can be used as a noun, properly called a nominalized adjective.
He did the unforgivable.
She achieved the impossible.
But it can also be used to describe people. Examples include:
The deceased is survived by his wife.
The blind use assistive technologies.
He's only trying to help the rich. He doesn't care about the poor.
Let's look at these sentences again:
She achieved the impossible (task of...).
The deceased (person) is survived by his wife.
He's only trying to help the rich. (metonymy)
The poor hope to live a better life. (metonymy)
The question asked is "when did 'impossible' become a noun," but perhaps it should be when did impossible stop being a noun?
As has been mentioned, impossible can be a true noun if we defer to the authority of the OED. Definition B.1. describes it as the equivalent of the noun inflection, "impossibility," most often used in plural.
The nature of an impossible becomes known from the seventh [theorem].
- 1789 T. Taylor tr. Proclus Philos. & Math. Comm. II. 6
Heaven sometimes converts our impossibles and inevitables into the very best blessings we have.
- 1866 D. M. Mulock Christian's Mistake 130
But this sense of "impossible" seems to have fallen out of favor. In its place, OED offers B.2., used with definite article, which, as has been discussed in KumaAra's answer, resembles a nominalized adjective.
This pattern seems similar to the use of "infinite," which also had a more prevalent noun form that is somewhat archaic. This citation from OED places the noun and adjectival forms in tandem:
Since every part of an Infinite is infinite, there may be supposed something more infinite than an Infinite.
- 1712 tr. H. More Scholia Antidote Atheism 151 in H. More Coll. Philos. Writings (ed. 4)
In both cases, the noun use likely grew out of the nominalized adjective, as the adjectival uses are attested first, and the semantic natures of the words lend themselves to nominalization.
The oldest example of "impossible" as a noun in the Oxford English Dictionary is from 1374
That wyst he wel an inpossible were.
--Chaucer, Troilus & Criseyde
and the oldest "impossible" as adjective is 1340. But of course either or both may have existed before that, but not survived in written form.