"Christ has conquered all of hades, and given all its inhabitants the opportunity to arise out of it to the Paradise of the living."

Agreed. Apokatastasis.

"And I say also unto thee, That thou art Peter, and upon this rock I will build my church [legislature, ekklesia]; and the gates of hell [hades, sheol] shall not prevail against it."

Notice the gates of the heavenly city in Paradise are always open, no more to be shut:

"And had a wall great and high, and had twelve gates, and at the gates twelve angels, and names written thereon, which are the names of the twelve tribes of the children of Israel: ... And the nations of them which are saved shall walk in the light of it: and the kings of the earth do bring their glory and honour into it. And the gates of it shall not be shut at all by day: for there shall be no night there."