OP: What documents during and before the first century use the expression "second death," and what did it mean? Did it reference (1) eternal torment; (2) permanent, literal death/annihilation; (3) temporary, corrective punishment; or (4) something else? (I'm asking specifically for extra-biblical sources during and before the first century, not sources after.)
Philo a contemporary who wrote and lived about the time of Christ describes a "second death" in contrast to an "ending death" with hope.
Perhaps some one will say he [Cain] should have been put to death at once; this is a human mode of reasoning, fit for one who does not consider the great tribunal of all for men look upon death as the extreme limit of all punishments, but in the view of the divine tribunal it is scarcely the beginning of them.
On Rewards and Punishments
For Christians, the great tribunal would be the great white throne judgment, which precedes the decision about the second death.
And I saw a great white throne and the one sitting on it. The earth and sky fled from his presence, but they found no place to hide. Rev 20:11
which then leads to judgment and the second death
Then death and the grave were thrown into the lake of fire. This lake of fire is the second death. Rev 20:14
This sequence is likewise with Philo.
Since then the action of this man [Cain] was a novel one, it was necessary that a novel punishment should be devised for him; and what was it? That he should live continually dying, and that he should in a manner endure an undying and never ending death; for there are two kinds of death; the one that of being dead, which is either good or else a matter of indifference; the other that of dying, which is in every respect an evil; and the more protracted the dying the more intolerable the evil.
Live continually dying, as in the lake of fire, without hope.
The word used for "eternal punishment" in Matthew 25:46 is kolasis [kovlasi"]. According to Bauer writers during the New Testament period used it only of temporal torture and conscious torment in the afterlife. No other idea for koine Greek is recognized. Moulton and Milligan can find only examples in papyrus where kolasis [kovlasi"] involves the person actually feeling the punishment. It is used elsewhere in the New Testament only in 1 John 4:18, which says fear has torment.
About 150 years later, Irenaeus picks up this theme of judgment and eternal destiny.
And in the Apocalypse John saw this new [Jerusalem] descending upon the new earth.4773 For after the times of the kingdom, he says, “I saw a great white throne, and Him who sat upon it, from whose face the earth fled away, and the heavens; and there was no more place for them.”4774 And he sets forth, too, the things connected with the general resurrection and the judgment, mentioning “the dead, great and small.” “The sea,” he says, “gave up the dead which it had in it, and death and hell delivered up the dead that they contained; and the books were opened. Moreover,” he says, “the book of life was opened, and the dead were judged out of those things that were written in the books, according to their works; and death and hell were sent into the lake of fire, the second death.”4775 Now this is what is called Gehenna, which the Lord styled eternal fire.4776 “And if any one,” it is said, “was not found written in the book of life, he was sent into the lake of fire.”
Irenaeus Against Heresies Book V Chapter XXXV
This quote from Irenaeus could not be any clearer that the lake of fire the second death is Gehenna.
Gehenna is found in the Gospels.
'And if thy foot may cause thee to stumble, cut it off; it is better for thee to enter into the life lame, than having the two feet to be cast to the gehenna, to the fire -- the unquenchable -- YLT Mark 9:45
In addition to Philo and Irenaeus, we find the same reference to judgment and the second death in the lake of fire with Tertullian, writing about the same time as the latter.
Accordingly, we get ourselves laughed at for proclaiming that God will one day judge the world. For, like us, the poets and philosophers set up a judgment-seat in the realms below. And if we threaten Gehenna, which is a reservoir of secret fire under the earth for purposes of punishment, we have in the same way derision heaped on us.
Tertullian Apology Chapter XLVII
The lake of fire the second death is termed Gehenna in the gospel accounts. This definition is found in Irenaeus. Further, Philo also writes of the great tribunal the great white throne judgment that may lead to eternal torment without hope in the lake of fire.