Suggestions:
Quezacotl: I'm not sure I'd go with Alchemist. I think some form of mage would better fit.
Shiva: I agree. Monk.
Ifrit: Warrior.
Siren: Why Black Mage too? I'd say White Mage only. Maybe you could make Quezacotl Black Mage.
Brothers: If it wasn't for the lack of magic, I'd say Paladin definitely, but I'm not quite sure. Geomancer is a magic class, and Brothers supplies way too much HP, so I don't agree with Geomancer from Brothers.
Diablos: Hm. I'd say Dark Knight, but I dnt think Diablos is high on the DEF scale. I think I'd go with Time Mage too.
Carbuncle: Definitely not Blue Mage (Uh... Quistis?) Not summoner either. I'd think some sort of specialized combat-based class, like Knight. Yeah, Knight sounds good.
Leviathan: Scholar. Black Mage + White Mage.
Pandemona: I'm not quite sure. I don't think Sorcerer would fit though. STR+, SPD+, Initiative, and Absorb don't quite craft a single class, I don't think. Absorb is what throws me off. If not for Absorb, I'd probably say like Black Belt or something like that, like an advanced Monk class, but Shiva pretty much has Monk covered.
Cerberus: I'd say Thief. Ninja would work well.
Alexander: Paladin works fine.
Doomtrain: Dark Knight ftw.
Bahamut: Eh, lots of mixed buffs here. My initial thought was Alchemist, tbh.
Jumbo Cactuar: SAMURAI SAMUAI SAMURAI (I was waiting for this one XD)
Tonberry King: I think like a cross between Corsair and Beast Master. Dunno really :S
Eden: I was thinking berserker, but only because that's all that comes to mind.
The only problem is that some of the classes would pretty much mirror other classes. Sounds interesting though.
Good luck!