I never said it did. The first death does not equal "non-existent" otherwise there would be no Resurrection and no Judgement Day.
But after the second death, those who are unable to overcome the hold of death will be no more since God will not raise them again. They will perish in their mortality. And so they will be separated from the Lord.
"Everyone does not have an immortal soul."
Yes, you've said this before too and I'll remind you again that you are free to say that, but you are not going to have a lot of supporters with that assertion since it doesn't make sense to say someone will exist forever but not be immortal and so most traditional hell view believers do advocate the belief that everyone, the wicked and the righteous both, have an immortal soul. And contrary to what you are saying, living for eternity in eternal torment, is still living. You can't live forever in a state of dying, but yet it be impossible for you to die, and that be equated with actual dying. Makes no sense.
"We are all spiritually dead before we get saved. Does that mean we were non-existent before we were saved?"
The key word there is "spiritually". We were physically alive before we were saved, therefore we existed, but we were not spiritually alive.