The newer 802.11n version of the AirPort Express offer an option to configure the base station as a wireless Ethernet bridge. Again, Apple calls this ProxySTA. You configure for a ProxySTA by having the base station join an existing wireless network AND by enabling the "Allow Ethernet Clients" option, located on the Wireless tab within the AirPort Utility.
My point is that, as a ProxySTA, the Express will NOT extend the wireless range of your D-Link router. Instead, it WILL allow wired clients to connect to its Ethernet port for network/Internet access.
In the manual setup settings, I see that there is a wireless setting to extend a wireless network and to participate in a WDS network. Would either of these two be better for what I'm trying to accomplish?
In this scenario, no ... as a WDS would not work because very few non-Apple routers are WDS-compatible. Most likely, your D-Link is not as well ... so your current configuration is the only option for what you are trying to do.
It seems that this current configuration should be rock solid, but perhaps I'm not using the proper setup.
The #1 problem with a ProxySTA is that it does not provide the highest level of bandwidth performance. Typically, for the AXn, the best-to-worst performance would be the following configurations: 1) Wired-to-wireless, 2) Wireless-to-wireless (WDS), and 3) Wireless-to-wired (ProxySTA)
As you can see, the first option would be the best, but it would require that you connect the AXn directly to the D-Link by Ethernet ... which, of course, would tie-up its only Ethernet port so that you could not connect the PC to it as you require.