Update billing address on open invoices:
The billing address when updated will not show up on the invoices that were generated already. The existing invoices can have the billing address updated as well. But this is not automatically done as these invoices are generated and sent already to the customer. If you would like to send an updated one you can use the action as shown in this video and send another pdf copy to the customer.
Automatically update card address on gateway based on the billing address updated on Chargebee site:
The billing address and card address are different entities and updating the billing address doesn't update the card address.
Also, if the address verification feature is enabled in Authorize.net or in any other gateway these payments fail if the billing address and card address don't match.