Of course, I know the Witch Cult intends to use Emilia's body as a vessel to resurrect Satella, but I wonder why? Maybe it's the fact she's a half-elf? Does the cult have other plans for her? Does the cult have some ways to contact Satella? And furthermore, did, and does, Satella agree with them?
Not revealed yet. But it could be a time travel/timeline story.
Each time Subaru dies, it’s actually a parallel timeline. Satella is Emilia from one of them... except in the future. Where Subaru did everything "right" but died.
Not knowing the truth, Emilia travelled back in time, so she can "save" Subaru from death by making him come back from death, unknowingly creating the situation leading to his death.
From the Wikia
Certain people, such as Echidna and Pandora, have referred to Emilia as "the witch's daughter", with Pandora also mentioning that she was of the witch's bloodline after seeing her power, hinting at some sort of connection between the two, though nothing has been explained.