With reference to Eden being female because it was a female's name - Shiva is the name of a hindu god - and well....shiva looks to posses pretty feminine features the last time i looked
So in my opinion, to decide on the gender...if any, of eden....i think you really need to decide what eden actually IS, and, well....unless someone's read something i haven't, i think thats pretty open to discussion. I don't really see eden as having a gender....just some kind of huge spiritual machine. Who knows? As for the others...(IMO)

Masculinity:- Quezacotl, Ifrit, Brothers (ha), Diablos, Tonberry, Cactuar, Bahamut, Alexander, Doomtrain, Cerberus, Odin/Gilga

Femininity: - Shiva, Siren, Carbuncle,

Undecided: - Eden, Leviathan, Pandemona