If I have one Airport Extreme set to "extend a wireless network" of another Airport Extreme, will it do so via the ethernet connection, if they are hard-wired together? Or will it connect wirelessly? Or, do I have a choice?
Both, I believe. All switches will extend each other when hardwired, but only some, like the airport, have the ability to also act as a wireless extender.
Like the above said. But if you're extending your network with two AirPorts, I really recommend you use WDS, and set your non-main AirPort as a "Remote Base Station".
It will all work nicely, with only "one" network visible to the clients.