Can time capsule operate with a wireless connection to the router?

Yes.

I assume that you are asking if the Time Capsule can connect to another router using wireless.....not using a wired Ethernet connection.

Yes.......IF......the Time Capsule connects to a wireless network that is being provided by another Apple router.

No......IF......the Time Capsule is trying to connect to the network provided by a third party router.

Yes

Providing you don't mind a slow network

Ted