The Pharoah's Hammer is actually Thor's HammerIt was dimensionally displaced, and the Fairies were charged with getting it back.
Doctor Impossible and CoreFire are the same personDr. I is already established as an Unreliable Narrator— he may also be delusional or have a split personality a la Fight Club. A lonely grad student thought up an imaginary friend— but his train-wreck self-esteem also made "Jason" something of a Jerk Jock, the kind of handsome, swaggeringly self-confident Chick Magnet and athelete he'd always been both jealous and contemptuous of. The Zeta Beam made his "popular" imaginary frienemy real and super-powerful, while also giving Impossible powers, which he suppressed at first. The second lab accident never happened— he just wanted a more dramatic mad-science origin, so his mind created the memory of bubbling chemicals and Tesla coils. He misremembers his high-school relationship with Erica/Lily— he was unable to accept that she just didn't like him that way and blamed imaginary Jason for "stealing" her. Doctor Impossible created CoreFire long before he was actually CoreFire. The pictures and history of young Jason/CoreFire, as shown at his funeral and in the Champions video, were created after the fact by Impossible, to support his delusion, and then "leaked" to the media. Impossible blanked his own memory of forging these documents. Necessary corollary: Dr. Impossible has memeory-altering powers and doesn;t know it, because Damsel, Lily and Blackwolf all remember "Jason" from school.
Lily's original backstory was the true oneAll of that business about being the local Lois Lane Captain Ersatz was just a cover story, to hide the fact that arranging the destruction of Pharoah's hammer, taking [CoreFire] out of circulation long enough for Kosmic Klaw to set a critical scheme in motion, disposing of Psychic Prime, and getting rid of Doctor Impossible's stockpile of advanced technology once and for all were the final steps in bringing about her apocalyptic future.
Baron Ether/Impossible's planned Ice Empire is actually the Empire On Ice.
Pharaoh's hammer is an Elfland relic.One, its an insanely powerful magical weapon. The only other weapon that come's close is the Elfland Scepter. Two, its cracked and losing power. The Scepter is also losing power. Three, during the Elfland story, the text mentions one of the brothers trying to hide his hammer when the rescuers show up. Also, the backstory for Pharaoh mentioned that he found the hammer in the woods when he was a boy. It all makes sense.
Elfland was the Land of the FairiesCombining the above guess with the implications that the Hammer of Ra was an ancient fairly relic that Elphin was supposed to get back.
The Peterson school was founded by Mr. Wammy.It is said that he founded such schools all over the world. It fits the criteria: A mysterious school taking in genius children for a nebulous purpose... check.
Dr. Impossible didn't give Corefire his powers. Corefire already had his powers, and they are what saved him from Dr. Impossible's experiment.Dr. Impossible has been seen to be a fairly unreliable narrator, and tends to skew previous events based on his own perspective. He regards Corefire as a dumb jock, but if Corefire was in the same high level classes as Dr. Impossible then he couldn't be as dumb as Impossible believes. In addition, he believed that Corefire clearly remembered him, but during their final confrontation Corefire had no idea who Impossible was (do you remember everyone from college?). And he clearly didn't recognize Lily as his constant abductee (who's own backstory is questionable at best). This shows that Impossible's and many others recollections of the past can't be fully trusted. In addition, it was stated that any attempt to recreate the experiment that gave Corefire his abilities failed utterly. Which means it might not have been random chance that allowed the energy beam to grant Corefire his powers. Corefire might have had his powers long before the accident, and Dr. Impossible wrongly attributes that event as the period which gave Corefire his abilities. But since he obviously thinks he's far more important to Corefire's personal life than he actually is, this simply would continue that theme.