Thankfully, I found the solution to the problem. A google search for specific ports for the AirPort Express lead me to this knowledge base article for all known TCP & UDP ports for Apple products:
The following two ports are specific to AirPlay:
554 TCP & UDP
Once I entered those two ports into the Port Forwarding section on the Motorola Cable Modem/Router, AirPlay suddenly allowed a connection to the speakers connected to the AirPort Express. The Motorola Modem is just awful to configure. My friend bought it because he didn't want a separate cable modem and Wi-Fi router.