I've tried plugging into multiple different LAN outlets throughout the house, and I am able to "Join an Existing Wireless Network," but I am not able to "Create a New Wireless Network" or "Extend an Existing Wireless Network."
This tells me that you likely have a problem in the Ethernet cabling or a bad port somewhere.
You can connect to a LAN port anywhere on the network, assuming that the Ethernet cabling and ports are all working correctly.
To test, temporarily turn off the wireless on a laptop and connect to the Ethernet port. If you can get a good Internet connection this way, then the cabling and port are working correctly, and can be used to connect to the AirPort base station.
If you cannot get an good Internet connection during this "test", you have a problem in the Ethernet cabling or an Ethernet port that will need to be addressed.
Once the Ethernet connections have been tested, you will need to start over from scratch on the AirPort and set it up again.
If you want a few tips on how to set this up, you did not provide the model number of the AirPort that you are using, or information about the operating system on the computer that you are using to configure the AirPort base station. We would need that info to provide correct instructions.