WiFi speeds slow down when AirPlay is enabled

When AirPlay is enabled, you are allowing the AirPlay protocol to "ride" on TCP/IP on your local network between the streaming host and the AirPlay "speaker." If you happen to be streaming, especially HD video content, while accessing the Internet, you will see an increased amount of data being transferred over the network which will take up the available bandwidth on that network. This could appear as lower download speeds from the Internet.

Is the Express your only router in your current network configuration? ... or is it either configured to extend another wireless router or to join its wireless network?

Just to confirm, since you recently bought the Express, is it model: A1392? ... and did you buy it new from Apple? What it the make & model of the Internet modem that you have it directly connected to by Ethernet?

Since your Express is connected directly to an Ethernet port, there must be another router upstream that controls Internet access to the building. That router would, most likely, be configured as such. That is, it will be providing both NAT & DHCP services for the building.

Did you reconfigure your Express as a bridge? Or are you getting a "double NAT" error message in the AirPort Utility?

Yes! This type of error occurs when you have two routers in series and this can affect both Internet upload and download speeds. However, if you reconfigure your Express as a bridge (which will solve the double NAT issue) you will "open" your local network to all other occupants in the building.

In theory, AirPlay should only be running on your local network even with a double NAT condition, so you still should be able to stream locally. Even without streaming being active, your computers will still try to verify that an AirPlay device is available on periodic basis. It shouldn't take up a lot of bandwidth doing so, so I'm not sure why you are getting the significant data transfer losses ... except for the double NAT issue.