How to stop the AirPort Express from showing up under AirPlay

You're connecting a router to a router, so you will get dual NAT.

The shared router is translating the single IP from the ISP into a shared network for you all to use. If you connect a second router to this to connect multiple devices, then you are creating a second network, and network address translation (NAT) is happening twice.

I'm not sire if the airport express supports bridged mode, but you could try this to see if it will allow you to connect multiple devices over Wi-Fi with the main shared router distributing the IP addresses. If you can do this, it should solve your dual NAT problem.

Dual NAT shouldn't however cause intermittent connection, that's more likely interference from other nearby Wi-Fi devices, so I'd suggest selecting different channels (if possible) until you find one that gives you a more stable connection.

It will work wirelessly if you are able to get Wifi from some of the old routers. In other words: you need Wifi signal from your existing network for both your AirPort Express and the Apple device to use AirPlay in this configuration.

The Airport Express will work on client mode and will not extend your network, but it will allow AirPlay. This is because AirPorts do not wirelessly extend non Apple networks. See this question for reference.

I have a similar setup in my home, and my solution was to run an Ethernet cable to the AExp to be able to use it to extend my network, and do AirPlay.

On the Error -6753, I've found someone suggesting you have to enable IPv6 on your router.