Apple uses the term " wireless extend" to mean that the AirPort Express receives a wireless signal from the main Apple router and then "re-broadcasts" or "re-transmits" or "repeats" that wireless signal again to provide additional wireless coverage.
This is a proprietary feature of Apple routers, so "wireless extend" will not work with other third party wireless routers. That is why nothing works when you try to setup the Express to "extend a wireless network".
If you have the 802.11n version of the AirPort Express......either the one-port A1264 or two-port A1392 version of the Airport Express......that product can usually "receive" a wireless signal from most wireless routers and convert that to an Ethernet signal. It has worked with every third party router that I've ever tried.
While you could call this "extend" if you want, Apple does not, since the AirPort Express "joins" the wireless network, it does not extend or re-broadcast the wireless signal from the main router.
Something is wrong if you have an A1264 or A1392 version of the AirPort Express since it can do what you ask....."join" a wireless network and provide an Ethernet signal for a remote device to connect. This connection though, will only be as good as the quality of the wireless connection between the AirPort Express and the main router.
In other words, you might use Ethernet to connect from your Mac to the AirPort Express, but the connection will be the same or worse, as if the Mac simply connected to the wireless network directly. There is no "boost" to the wireless signal, so using Ethernet to connect from the Express to your Mac just makes things more complicated, since there is some loss in the wireless to Ethernet conversion process.
You will likely have better results by simply connecting the Mac directly to the wireless network......assuming that the WiFi card is working on the Mac.
If you have an older A1084 or A1088 version of the Airport Express, that product cannot do what you want, so no sense in continuing the discussion regarding that product.