But this mercy is no longer possible after death, correct? Bascially, after the physical body has died, the soul cannot ask for mercy. Correct?
Correct.
The soul has made an eternal choice - for God or against God - and because God respects our free-will, He will not force us to love Him. So, even if a damned soul could be saved, God would not force them to repent, and because the damned’s wills are set in evil (just as the saved’s wills are set in good), they would not want nor accept God’s Mercy.
Now, to clarify on Purgatory: I said it was punishment, but also a gift from God’s Mercy. Now, how can the condemnation of a soul to suffering be called a gift of God’s Mercy? In itself, Purgatory is the work of God’s Justice: Mercy dose not want the soul to suffer, but Justice demands it. Some would say this makes God against Himself, but it is actually the opposite: in allowing the soul to be purged, in accordance to Justice, He is giving the soul a way to Heaven - rather than the soul just going to Hell for being stained with sin - and it is in this way that Purgatory is a work of Mercy. Those in Purgatory are already saved, precisely because they’ve chosen God and accepted His Mercy at death, and so God has assured them of Heaven, but they still have venial sin, so God wipes the stains of sin off their souls, making them holy. Just as He has given them eternal life, so now, in Purgatory, He gives them purification; just as the soul accepts eternal life, so now, in Purgaory, it accepts purification: though the soul still longs for God, and this is quite natural indeed, for we are made in His Image and Likeness, we will not rest until we rest in Him, we long for Him because He is our happiness, our life, and our everything: to not long for God is to cease to be His creature, so to speak.
Those who’s physical bodies have already died on the final judgement day would have already had their eternal judgement at the time of death. Why would they be judged again at the last judgement if they have already been judged at the time of physical death?
The general judgment question is an ancient question, and it is right to ask such questions, for in asking we come to a greater knowledge of God, and the more we know God, the more we love Him.
The reason for Judgment Day is not because God has prophesized it through His prophets, but because He must judge those who have not been judged yet when Christ comes in glory, and because He demands a proper estimate of the good and bad actions of all, and because He wills to show all that everything was ordered by an All-Wise and All-Just God. In judging men, God will reveal the good and bad of all, and in revealing the actions of all, He will show that everything was ordered by Him.
Those whom God has already judged - who have already gone through the particular judgment - will not be judged in the manner of either condemnation or reward (“You will go to hell!” “You will go to Heaven!” to use a human way of speaking), but rather, they will be judged in the manner that all will be judged on the last day: their good and bad actions will be made known to all. Indeed, the good and bad actions of every man will be revealed, including those whom not been judged yet when Christ comes in glory, and for these lot, they will be judged in the manner like all men are at particular judgment: either going to join the saved or the damned. Thus, when all the saved are assembled before Christ and all the damned are grouped together, the Lord will invite the former to their eternal reward and He will condemn the latter to their eternal punishment.