What does AirPlay need from wifi and router?
AirPlay uses the following ports:


TCP 80 - http
TCP 443 - https
TCP/UDP 554 - rstp
TCP 3689 - daap
UDB 5353 - mDNS


Data from https://support.apple.com/en-us/HT202944 - TCP and UDP ports used by Apple software products and TCP and UDP ports and protocols used by Apple TV

Here's how to test ports and also an AirPort centric setup guide