Sure there is a mention of this.
1. Satan is a thief.
2. God operates according to the law. That is why people will "sell their souls to the devil", it is a legal contract.
3. If a thief is caught stealing and found guilty in court then they must repay 4x or 5x depending on if it is a sheep or an ox.
It is reasonable to think that since Satan is a thief some of the souls he has were stolen and therefore he must repay not just the stolen ones but 4-5x as many souls.
Of course my reason passes your "karma" test. God is righteous and just, if someone chooses to live in a place without God then they can. But that means they live in a place without love, hope, faith, light, life, water, etc. Since our soul is eternal they will live there for eternity.
As for the whole punishment vs punishing, I consider that simply word play. God sends them to hell, but once in hell they are tormented because they realize what they gave up and how much God was good to them and they didn't appreciate it.
I see no Scripture cited to support your novel belief that souls get a second chance and go to a third place.
Souls are NOT eternal but were created and will not be destroyed after justice in hell is attained.
(John 17:12, Rom. 9:22, Gal. 6:8, Phil. 3:19, 2Thes. 1:9, 2Pet. 3:7 & Rev. 20:13-14).
Again, hell is eternal punishMENT, not eternal punishING.