First question: The law does not prohibit teachers from carrying in a school if they are Delaware residents with a Delaware CCDW, but the terms of their employment may say otherwise. If caught carrying, they may well lose their jobs, but would not be charged with a crime (unless their lawyer sucked).
Second question: Delaware residents with a Delaware CCDW can carry in a school, or on school property (but not in a school bus).
I've carried in a school a few times. It seemed more responsible than leaving a loaded gun in a car, and my lawyer's office was just up the street anyway...