In the Airport Delays Page (for a given airport), is the Airport Delay index related to the FAA Airport programs and FAA status?

No, not directly. The Airport Delay Index values are calculated using a proprietary FlightView algorithm based on the departing[arriving] flights in the past 2 hours, and how late they departed[arrived] relative to their planned departure[arrival] times. The FAA Airport programs and FAA status are defined and issued by the FAA (Federal Aviation Administration).