On why not just kill the baby secret keeper:
Well, there's the Circle of Protection and then she's hidden among Muggles. Plus I expect there are laws against just offing an innocent child (if you get caught, but those who knew where and who Alexandra was probably also knew that Claudia is aware of the wizardig world and might raise an outcry there, even if Abraham Thorn himself didn't ...). I imagine a lot of people were hoping somebody else would do it.