When can I disconnect the cord and have it work wirelessly.
Sorry, but this is not possible.......unless......you have another Apple router producing the wireless network. Might that be the case?
how can it be an extension of my current home network, rather than a new network?
It can't, if you have another brand of router and you are asking about having the Time Capsule connect wirelessly.
However, if this is the case, you can move the Time Capsule to the location where it will provide additional wireless signal coverage and configure it to create a network that uses the exact same wireless network name and password as your "main" network.
You can locate the Time Capsule anywhere that an Ethernet cable will reach.....up to about 330 feet....or 100 meters.