Can an Airport Express be used to extend the Wi-Fi of another-non Apple Access Point?
Apple seems to be saying in a round-a-bout way here that other routers are not supported for wireless range extensions (with a router connected to Ethernet and the Express not):

Extend an existing Wi-Fi network’s range.
If you already have a
wireless network in your home and want to extend its range, AirPort
Express can help. Just place it in range of your primary base station
— an AirPort Extreme, Time Capsule, or another AirPort Express — and
near the area where you want your wireless connection. Launch the
easy-to-use AirPort Utility app on your iOS device or Mac, and you’re
mere minutes away from long-range Wi-Fi enjoyment.

However, this is only if your AirPort Express is not connected to Ethernet.  Otherwise, you can setup a roaming network. The newer Airport Express models (the ones compatible with 802.11n) can do this.

First, you need the Express connected to Ethernet.  Make sure that the main non-Apple router is set to share a public IP address.  Next, when setting up the Express, just set up its Network name and Password exactly the same as the network used by your non-Apple router.  The network mode needs to be &quot;Create a wireless network,&quot; it should Connect Using &quot;Ethernet,&quot; and Connection Sharing needs to be &quot;Off (Bridge mode).&quot;