I think we should be able to receive notifications in the Stack Exchange iOS app when we get/lose reputation or earn badges. We could be able to specify in the app if we want to be notified in Notification Center when one of our questions/answers was upvoted/downvoted and other changes or just be notified when somebody comments on our answers/questions or answers our questions. Is anyone capable of putting this into the app?
I don't think there is really much benefit from adding that. In my opinion, notifications are there to inform you some action should be taken. Reputation and badges don't require any action.
According to this somewhat similar question about email notifications for badges and reputation, SE doesn't want to have an extensive list of options to toggle on and off, like you propose. That would mean this is a all or nothing implementation, and I don't think all users want such an option.