Does Satan rule Sheol?

No. The spirit world, even spirit prison, where Jesus taught the spirits who died before Noah, is not controlled by Satan but is a place without glory where the inmates are faced with their own shortcomings, where they can feel the remorse for the wrongs they committed. Satan does not have to be there for the spirits to realize that they were wrong.

Satan only rules over the Sons of Perdition, or what we call Outer Darkness where the souls of resurrected people go who received no forgiveness for their sins. Only those who sin against the Holy Ghost, denying Christ after knowing the truth, will be with Satan in His domain. That is different from Spirit Prison.

Yes, it is possible that he does.

Even Christ admit that (John 12:31), as Satan was a lucifer before Jesus overtake that title, Satan ruled all elements and was the prince of all realms (life and death) world, including the realms of the living and the dead. The strong references goes as

The Bible stated God is the God of The living but not the dead, but the whereabouts of spirits of both saints and the mere afterlife were unclear. But as Christ told us in John 12:31, and Matthew 4 shows Satan s unlimited rulings. It is very possible that Satan rule all worldly realm prior to Christ Messianic Days. This is logic that Satan sought after either to make Christ fail to His Messianic duty or die.

The saying that God is the God of The Living proven men should not die, as Enoch, Moshe, and Eliyahu stand before The Father s presence, taken up alive, as God trying to show that originally that He never meant men to die, however it is possible that these elders later accused by The Accuser before The Father in Heaven in Revelation 12.

Other debatable case also Samuel s ghost whose appeared before Saul (who against The Law sought after ghostly counsel) in 1 Samuel 28 and even though Abraham erected Ton Kolpon at Sheol/ Hades, he too was suppressed under Satan s rule, until he saw the day Lord Jesus is about to come (John 8:56).

No cause Satan and these f... Demons that the Romans took from pagan myths don't really exist... from what I know wouldn't believing in Satan be a demi God or any other demon, Also there are different types of Paganism, Nordic pagans didn't believe in demons or the devil their never was such a thing to them in till Christians took over and forced their religion on Nordic pagans

Satan rules nothing according to Christianity. If I recall, he was a prisoner

You Betcha!

He certainly rules ISIS and HAMAS!

"Sheol" ( שְׁאוֹל ) simply means "under the ground" or "under the dirt"....so to answer your question; no....