I highly doubt there would be a bank for bitcoins because the whole point of btc is for it to be anonymous and that bank will totally destroy that ideal.
I don't think so, many bank nowadays accept cryptocurrency. Besides if you want to cash out your earnings you will get the money in the bank. Blockchain is a good way to communicate because aside of its privacy it is also used in transacking a money without the approval of many. Bitcoin might close the banks but they could not stop it.