DD did have nightmares from book II-- just so that you're aware. I think that much depends on a child's awareness and sensitivities. Ginny's (effective) possession in book II was something that my DD found extremely disturbing.

There is pretty good plot summary at the wikipedia page.

http://en.wikipedia.org/wiki/Harry_Potter


The third book includes an adult that 'stalks' Harry, and the introduction of the Dementors.



As I said, though, I think that most children probably won't pick up on those super-scaring foreshadowing things. DD did in some places. I really recommend reading the series WITH a gifted child so that they can talk with you as they go. Some of the scariest things aren't "magic" things at all-- they are the very human (and dark) motivations and actions of the supporting characters, like Barty Crouch in book III, Severus Snape, even Draco Malfoy (and his family). There is racism-- and it's potent.

Schrödinger's cat walks into a bar. And doesn't.