On-demand deposits will be stored at the bank, and will not be lent out. You will likely pay a small fee for this service.
In this case the bank might just store your encrypted wallet to ensure you that your money won't be lent out. Or you could just maintain the backup yourself without a bank.
Yep, but then you couldn't take advantage of remote checking services unless you provided the password to decrypt the wallet. But that's no different than just giving them the coins.
I fully expect that both types of services will exist.