Q: Do I need to constantly have my airport express connected to my home router for it to act as a wireless airplay receiver

The express can be used on it's own, however:

If the base router is not an Apple Airport Extreme, then the Airport express cannot act as a wifi extender for it, and will have no internet access, (yellow light will always be flashing) nor any connection to the network that is connected to the base router. In this case the Airport express must be connected to the base router via ethernet cable always.

Devices and computers may still be able to connect to it and stream to it, but will have no internet access.

If the base router is an Airport Extreme the Express can extend the Wifi network.

Wi-Fi base stations: Extending the range of your wireless network by adding additional Wi-Fi base stations - Apple Suppo…