Does this work with a non-apple router, such as the TP-Link TL-WR1043ND?

I own a TP-Link TL-WR1043ND and I wondered if the same function can be accessed, using the Time Capsule to extend the TP-Link's network.

The Time Capsule can "extend" a network from another router if you connect it back to the main router using an Ethernet cable.

Configure the Time Capsule to "create" a wireless network using the same wireless network name, same security, and same password as the main router. Make sure that the Time Capsule is in Bridge Mode and it will in effect, "extend" the network that way.

"Extending" using wireless only was designed as a proprietary feature of Apple to Apple products. It is extremely unlikely that it would work with a non-Apple product.

do I simply configure the TC to "Extend a Wireless Network", or do I Create One, but of the same name as the one on the TP-Link?

With an Ethernet connection, you would choose the "create a wireless network" setting using AirPort Utility. Things can get tricky because the network must be configured to use the exact same name, same wireless security and same password as the TP-Link network. Bridge Mode is required.

if I picked up an Airport Express and used that as the principle router instead of the TP-link, would that solve it wirelessly?

Yes. 2nd Generation AirPort Express will certainly work.