[@Emma] For easy reference, I'm answering everything in bullet points! If you need clarification, let me know![list]
[*]I'm afraid your idea won't work in this roleplay, as interesting as it is. Hades and Lucifer are not interchangeable here. Lucifer is an archangel, and Hades is a god. In terms of power and what they are, they're vastly different in this roleplay.
[*]Lucifer and the other archangels never ruled the kingdoms that hold their names - the kingdoms were just named in their honour.
[*]The ruler of the kingdom of Lucifer specifically rules by power - it's a monarchy not ruled by family line, but by whoever fights their way to the top. Sometimes, though, a family line can be powerful enough to hold the throne for an extended period of time.
[*]For future reference, children of demons and angels work like this; their children are always human. A human child can be possessed by either an angel or a demon, but the child of an angel or demon will always be human, regardless of their parentage. That human child can also be either blessed or cursed by an angel or demon.[/list]
I was thinking - if you really want to have a character that's the child of a pair like Hades and Persephone, you could have a demon and a cursed/blessed human that had a child, who would be human, but then was either cursed or possessed by a demon, or maybe even blessed by an angel. It's always interesting to consider a demon parent cursing their own child, though. With that power, she'd be able to hold the throne in Lucifer, as long as whatever her power/ability is happens to be able to be weaponized.
(Her parents could even be the former king and queen of the kingdom - why they would give up the throne is up to you.)