Hey,
The cost/benefit wording on Satan's Seat is just a different, and less elegant way or stating what is addressed by the clarification in the place entry of the REG.
An ongoing ability paired with a place ability has an implied condition of “while the placed card remains in the placed location.”
When the site is taken, Satan's Seat follows the site and remains on it. As long as no other card is used to affect the placement or location of Satan's Seat the negate and discard abilities continue to work.
Professoralstad seems to have forgotten the precedent set with Magic Charms that if you take control of an underlying card from an opponent, you do not take control of any cards placed on it. Satan's Seat is now in the opponent's territory, but it is still controlled by the player that originally placed it.
The negate ability has nothing to do with the site that Satan's Seat is on. It is "directed" to the opponent of whoever is carrying out the ability, and the player that originally placed it is still carrying it out. So the negate continues to work as if Satan's Seat hadn't been moved. The discard ability cares only about the site that Satan's Seat is on, and not about who is rescuing from or defending that site. So the discard ability now triggers whenever the player that took the site surrenders a lost soul out of it, but the discard is still carried out by the player that originally placed Satan's Seat.
So if I play Satan's Seat on my site, you take the site that Satan's Seat is on and then put a lost soul into that site. If I attack you and you give me the lost soul out of the site with Satan's Seat on it, I discard the heroes I rescued with as a mandatory discard action triggered by my Satan's Seat.
Tschow,
Tim "Sir Nobody" Maly