Q: Airport Time Capsule : extension to existing network

What is the make & model of the wireless router that you want to extend with the Time Capsule (TC)?

Sorry, but you won't be able to, at least, not via a wireless connection between them. If, however; that connection can be wired then you can configure them both for a "roaming" network.

Not if you want to connect the Time Capsule using wireless. It must connect using a permanent wired Ethernet connection.

The Time Capsule would only extend/work using wireless if you had another Apple router....Not a BT router.

Sorry, but most likely not. The reason is that Apple uses proprietary technology for wireless extension that works best with just their own products. This is not so much different than with the other router manufacturers.