I think you should be patient on this one. I think this may be a different perspective than you believe.
I agree that hate is everywhere and your comment about Nightcrawler is absolutely spot on....but it's also nowhere, if you don't let it in and feed it.
Yeah, that's as clear as mud, I know.
Anyway, as I understand this opening story line so far, hate should be Hate, as in a name. I think in this arc, Hate is an external entity and, from that point of view, no, Hate cannot reach the child where they are in Atlantis. (I also think that phrase will come to have multiple and complex meanings, as the story progresses.)
I'm guessing that Hate is taking control of people and making them do things they wouldn't ordinarily do, and that Jean, as a telepath, has picked up on it, and has the team following its trail. She's taken the girl to their base to get her out of its sphere of influence temporarily, until it can be ferreted out and dealt with. That's what I think, anyway.
I'm expecting that one of the things that may come from this is the little girl learning that Hate, the villain, may be far less scary than hate in the world, because the bad guy can be defeated much easier, but who knows.
The opening scenes are reminiscent of
God Loves, Man Kills, though, so that is giving me hope that Taylor is taking the X-Men back to their genesis in some way, rather than the last decades of public faces and PR.