Some pedophiles hate being pedophiles.

Normal? No, it's not 'the norm' but there's a difference between normal and naturally occurring. I don't have enough knowledge to say whether it's naturally occurring or due to experiences, but I would stick my neck out and say it's not 'chosen'.

I'm not getting into the morals of it. To me, actions, not urges, are to be judged morally, so if a person has certain urges but doesn't act upon them and causes no harm to others, 'moral' isn't an issue. The argument could be made that a person who chooses not to act upon their urges because they know it's wrong is acting in a moral way.