The Trojan war was proven by, when people found evidence like armor and journals.
They built a wooden trojan horse
Achilles was killed before the sacking of Troy, there was no "after the Trojan War" for him.
Well, people are still debating, because we still have no historical evidence that it happened. But there is evidence that a fight did break out. So we're still not sure about it.
The best guess is in the Iliad; you might however look into the archeological evidence at the Troy site.
It didn't exist then.
They went home.
Nothing
He was killed with a knife from Agamemnon.
No, the volcano that wiped Pompeii off the map took place long after when the Trojan War is thought to have happened.
Nothing. Virgil was a Roman writer who wrote about the events following the Trojan War, specifically the journey of Aeneas (a Trojan hero who barely escaped the burning of his city and eventually went on to found a fairly unremarkable city called Rome). Virgil was writing centuries after the Trojan War, so nothing happened to him during said war.
He was the leader who led the Trojan war, later on killed by his wife(not really mentioned in Greek mythology)
The Trojan War happened near the end of the Greek Bronze Age, sometime between 1260 and 1884 BC.