I really doubt the source of power is the necklace. But rather her species
Uh... I'm not sure really, at the very least it amplify her power but I still think it give her the powers, same goes for Miller honestly.
They only look demon-like with the necklace on them. Alice did get beaten up by Agetha too with her having the necklace.
Alice existed before fpe and I never saw her have powers outside fpe series
