In the Airport Delay Map, are the delays (red, yellow, green dots) in the map related to the Airport Delay Index for Departures on the Airport Delays Page?

They are related. They are both based on flights that have departed in the past 2 hours. The delay dot in the Airport Delay Map is an overall measure of delay for all these departure flights. The Delay Index is a breakdown of these flights into the 3 index categories ('On Time', 'Late', and 'Very Late').