They'll accept bitcoins for deposit, with a very small interest. They will also issue loans for bitcoins, with a much bigger interest rate. So they make money on the spread.
Just like any other regular bank account, but in 3rd party currency. You'll keep your money with them because if you don't do anything, it's supposed to earn a little interest.
The rate would be very small, but it's got to be higher than zero otherwise you'd just keep it in cold storage.
Would not work. The banks would have no protection against a run on the bank. There is no central bank of Bitcoin to give out emergency overnight loans etc. CDs would be legit though.