The block chain is just newer ledger technology, there is no reason for banks to be unable to incorporate concepts based on bitcoin.
Yes, but what they would have would be a an intranet ledger for their own use. And if it has no token of worth attached to it then it's simply the same as programming whatever you like into an intranet ledger. That's not really a blockchain. That's just a cheaper banking system (for them.)
Huh, the banks would use it for the Dollar as an internal ledger, yes. Using block chain technology. They could develop that into many things. If your definition of blockchain is "not a blockchain if there's no cryptocoin attached" sure, but that's not what the blockchain is.
You don't need bitcoin to generate blocks of transactions and have internal nodes accept/reject.
A centralized blockchain is nonsense, it doesn't do all the cool features that a worldwide decentralized blockchain does. A centralized blockchain is an huge target for hackers and will never be even 1% as safe as a worldwide decentralized blockchain, specially the Bitcoin one which is backed by the most computing power ever seen.