Does AirPort Express require a wired connection.......if I already have a wired AirPort Time Capsule on the Network?
No, but you will have much better network performance if you do wire the AirPort Express.
The reason for this is that when you try to extend a network using wireless, you will typically lose about a half of the potential speed capability on the network.
There is no network speed loss when you wire the Express.
Here is what Apple's support document says about this:
In the case of a wirelessly extended network, throughput may be reduced to less than 60 percent of that of a single device. The general rule is to keep the Wi-Fi network as simple as possible. You can accomplish this by using the minimum number of Wi-Fi base stations required to service the physical network area and by using Ethernet wherever possible.
Extending the range of your Wi-Fi network by connecting Wi-Fi base stations together using Ethernet is always the best option, and will provide the best throughput