I have zero trust in my bank but every transaction will not need to be verified throughout history.
No, that's not true. You have 100% trust in your bank to hold your money for you. The bank says "you have this amount of money", and you believe that number. You don't control the money, the bank does.
On the other hand, Bitcoin *is* zero-trust. If you run Bitcoin Core, you're validating every piece of transaction information that you receive, and not trusting or believing anything just because someone tells you it's correct.
No, you're completely incorrect. A bank tells YOU that you have that amount of money and YOU believe that number. I do not.
Bank records and statements are accounting methods that need to be checked against your own records. If you're doing what I hope your mommy and daddy taught you when they helped you set up that shiny new first bank account, you are keeping a record of transactions (money you've spent and deposited) and reconciling that against the statement you receive from the bank each month. There is a tool I like to call a "checkbook register" to help you perform this task. That will ensure that the bank is keeping accurate records of your account and allow you to correct errors with the bank should you find an error or fraud against your account (zero trust). I have found errors in my account in the past, challenged my bank and they immediately corrected the error. If they fail to correct the error and you know they are wrong (because you don't trust them and keep your own record) you can still have the error corrected by using another tool I like to call "an attorney" to correct the error. Here is a fairly cheap book that can help you with a basic understanding of these concepts:
http://www.barnesandnoble.com/listing/2691051713384?r=1&cm_mmca2=pla&cm_mmc=GooglePLA-_-Book_15To24-_-Q000000633-_-2691051713384Record keeping should always be questioned even in Bitcoinland. However, there is a point in record keeping when the information (check register) is no longer valuable. That point is after reconciliation when the transaction is complete, it has been checked by both parties, those parties are in agreement and the information is not required for audit. It is then safe to dispose of your ledger and move on. I opened my first bank account (with the help of my mommy and daddy) in 1968. I no longer keep records of my transactions for that account because I would like to store automobiles in my garage instead of boxes of ledgers and they are unnecessary.
Bitcoinland banks which are uninsured and must be trusted completely are called "Wallet Services": inputs.io, MyBitcoin, Blockchain.info etc. Online banks that are insured to $250,000 of loss and deal with fiat are just called "banks" and include: Capital One 360, USAA, Flagstar, ally etc.