All Airport devices can join a network using Ethernet, and provide a WiFi hotspot in bridge mode (vs router mode).
All Airport Expresses should be able to join a WiFi as a client to connect airplay devices.
AirPort Extreme\TimeCapsule can join a WiFi or ethernet to share Time Machine disks.
I am doing all of the above on a Synology router network.
The only thing that might be a problem is wirelessly extending a network, that is using WiFi to connect the Airport to a network, then providing hotspot for client connections. In my experience, some routers need to designate access points to extend the network, and this may or may not support other vendor products (I ran into this many years ago with Belkin routers in the 802.11g days).
So, I would expect you can achieve your goal with just about any router. The Express should be setup in Client Mode on the network tab, this allows it to join the WiFi as a wireless client, just like a smartphone, tablet, or laptop. You may have to enable UPNP on the router, some don't natively support the bonjour protocol that Airplay uses to discover airplay devices. You would have to read the router manual, or forums, but I suspect most modern routers support bonjour out of the box given the popularity of Apple products.