The fact that the TC is possibly not operating correctly may affect how it "extends" the signal.
I would turn off the AirPort Express for now, and work on getting the TC working correctly.
Start with a power down, wait a few minutes, then power back up.
Check to make sure that your laptop can connect using Ethernet to the TC with wireless turned off on the laptop.
If the Ethernet connection still does not work when you test this, you may have a bad Ethernet port on the laptop, or Ethernet is not configured correctly on the laptop. That would explain why you cannot connect using Ethernet on the laptop with both the TC and the Express.
Here, I am assuming that the TC is acting as the router on your network and the Ethernet ports are working. Test another device using Ethernet to confirm if you need to.
If the Ethernet connection works correctly with wireless turned off on the laptop, then disconnect the Ethernet cable and test the laptop using wireless to make sure that it is working correctly with the wireless signal that the TC provides.
If that test works, then you can re-check the AirPort Express again to make sure that it has been setup to "Extend" a wireless network.
If yes, and the Ethernet connection still does not work, I have to conclude that you have a bad Ethernet port on the AirPort Express.