Mana Khemia: Alchemists of Al-Revis/WMG

Jess is descended from Bergholt Stuttley Johnson
Jess is one of the more incompetent alchemists, when it comes to synthesizing at least, in the school. Her medicine is near lethal, and she has been known to create explosions when working with ingredients that simply should not make explosives. Johnson is similar, in that he is recognized in Discworld as the only person who could create an explosive out of sand and water, an ability which Jess must have either learned or inherited from him. But how did Johnson get to the world of Mana Khemia? Easy, he's "Bloody Stupid" Johnson. This is a man who made pi equal 3 resulting in a mail sorting machine spitting out letters from various time periods including the future. If anybody could accidentally create a portal to another world, it's him.
 * He made a TARDIS through complete accident, with no involvement by any Time Lords, and installed it into a sapient pearwood casing. It is now Jess's handbag.

Flay is Kamina from a different continuity.
Flay is Hot-Blooded, charges into situations without thinking, makes a lot of statements about manliness to Vayne, who he perceives like a kid brother, and, biggest tell of all, his best attacks turn his sword into a drill. Mana Khemia, therefore, must be the TTGL universe that uses magic instead of science. Furthermore, the Mana must be like the Gurrens and Vayne's wish powers are caused by Anti-Spiral forces.
 * But that means Vayne would be Simon and Nia at once - *head explodes*
 * Well, when you think about it, there is both Vayne and Other Vayne, and Other Vayne fades away at the end. So would Vayne be Simon and Other Vayne would be Nia?

Vayne is a resurrected corpse.
All the townsfolk are afraid of Vayne whenever he goes into town, and Zeppel assumes that this is because they don't understand his abilities. Zeppel's referring to alchemy here since he'd have no idea about Vayne's other powers, but Vayne mentions that he's never heard of the word until Zeppel mentioned it, and at school he even worries about whether or not he'll be able to keep up, meaning that he's never done it before whether or not he knew about the word itself. It's possible that Theofratus was batshit insane enough at the time he made Vayne to have dug up some kid's corpse and infuse it with Mana powers or something. All that fear was because the townspeople swear they saw Vayne die. Incidentally, if he's a human body infused with the power of a Mana, that explains how he as a(n artificial) Mana could bleed, seeing as how normal Mana are apparently pure elemental power beyond their physical forms. ...And it explains how losing his powers just turns him into a normal human.

Vayne is Haruhi Suzumiya.
...someone had to say it.

Melanie is actually a vampire.
The game pretty much dropped this plotline, but it explains the dripped blood, and why she wants students to volunteer themselves to be unconscious for a few days. The joke was that she wanted them to volunteer because she's a pervert or something, but she never specified anything beyond several students to volunteer.

Vayne's final wish was
He didn't want his powers anymore. Makes sense.

Vayne's final wish was for there to be no more Mana
However, just like when Scarlet Witch tried it with mutants, it didn't quite stick. As a result, fifteen years later, Mana are extremely rare and almost extinct, but not quite.
 * Disproved by the sequel. Manas aren't going extinct, the Light Mana is forcing them to return to the Mana Dimension. All because

Jess's mana is not a mana of wind, but a mana of Chlorine Trifluoride.
Its power leaks all over Jess's attempts at alchemy, which is how she manages to make everything blow up. Her "wind"-based attacks are powered by the toxic chlorine gas released when chlorine trifluoride goes 'splodey; this gas is stored in the freakish inexplicable pocket dimension inside her handbag until needed for use.

Jess's handbag is a larval Eldritch Abomination.
It blatantly ignores physics and can grow teeth and chew stuff to death.
 * A scene implies that it may be the spaceship of one of Muppy's relatives. Of course, it could be a Living Ship Eldritch Abomination... That would explain what happened to that relative, after all.

In the sequel, Pepperoni is a descendant of Poe from Atelier Iris 2: The Azoth of Destiny.
By the end of Atelier Iris 2, "Flashbang" Poe had (semi-unwillingly) become married to a Catgirl. This is, to this troper's knowledge, the only successful relationship between a fairy and a humanoid species. And, although Pepperoni doesn't show any cat-like qualities, he's certainly not a purebread fairy/wood elemental either...
 * The Mana of Light's excuse for everything is that a mana died giving birth to a human's child. That child is most likely Pepperoni.

Nikki is a descendant of Norn from Atelier Iris.
Catgirls. The same might be said of Repre of Atelier Iris 3: Iris notes that theirs is a rare species, which makes sense as Norn wasn't a normal beastman but a mountain cat turned into one by Zedalia.

Flay is the child of...
Justi Ueki Tylor and Empress Azalyn via test tube. He's trying to learn to play people like his father, the world conquering thing comes from his mother's side of the family and the OVAs show Azalyn using earth television as a guide to behavior...