Lol, lots of questions there. Lets see what can be answered:
So is Leah superpowered just from being Diablo's daughter, or was she just a shell for another copy of Diablo?
She's probably absorbed some power from being Diablo's daughter, and she is used as Diablo's host because she's more susceptible to him because she's born of Diablo's blood. She didn't start out being Diablo's shell, but became a shell in accordance to Adria and Diablo's plans.
Did she have to be the one for the soulstone transfer thing to work, or could it have been anyone?
She didn't have to be the one, but again, since she's of his blood, she's probably more susceptible to Diablo's influences.
As well, why does everyone think Diablo is dead at the end?
Diablo is indeed 'dead.' His physical shell was killed, but his essence always lives. All the primary angels and demons are immortal, and return to the Abyss (or Arch) to be reborn after some time.
Is that because the stone only holds the power of the lords, and that Leah was Diablo all along, or did everyone forget about how the soulstone works?
The Black Soulstone is an anomaly in that it was fashioned by Zoltun Kulle. We are not sure how its properties work, but it is assumed that it absorbs the souls of fallen greater demons. Hence, Diablo's soul has returned to the black soulstone.
Leah was not Diablo all along. Adria uses her magic to 'encourage' Diablo's will to take control of Leah's body.