at is a bank obligated to do if a customer notices that their account has either more or less money than it should and the customer has not deposited or withdrawn any money?

OK, so this actually sounds like my area of expertise. If this is a deposit account, it is governed by Reg E. So we'll start with "OK, your account magically can't have less money than it should without something happening to it." A transaction must have occurred at some point that caused the balance to be off. Examples:

  - you may have tried to deposit $200. When you check your statement, for some reason, only $100 is reflected (Numbers purely for example purposes) 

  - There may be a transaction on your account you did not authorize (aka, someone took money out of your account, you don't recognize it and you didn't say it was ok) 

  - You may have tried to withdraw a certain amount, let's say $200, and the ATM only gave you $100, but it says $200 on your account statement 

Somehow, there was a transaction that occurred on your account that caused the balance to reflect (as you would think) incorrectly. Luckily, consumer, you have recourse available to you! Here are the steps to take:

  1) Find the transaction on your statements. Hopefully find it within *60 days* of the *statement* on which the error occurs. 

  2) Call your bank. If a merchant committed the transaction the bank will tell you to call the merchant. You will, and you will ask for your money back. If you are lucky, they'll give it to you. If they won't...

  3) Call your bank back. At this point you will be initiating a *Reg E Error Resolution Process.* Your bank now has 10 days to investigate your claim and make a decision. If they need more time, they can take 45-90 more days (depending on transaction type) but *they have to give you your money.* Also, if you incurred any fees because of this transaction (like an overdraft) *they have to refund it.* 

  4) Now your bank is going to look at the transaction, reach out to the merchant, and serve them up some paperwork saying "You owe us money." This is called a *chargeback.* If the merchant disagrees (and most of the time they just kind of go with it) they will say "Nu-uh no we don't" and send your bank a rebuttal document called a *Representment.* The bank will either say "This isn't a valid rebuttal" or then go back to *you*, the cardholder, asking if you can provide any more information. If you can provide proof the charge isn't valid your bank will file a *Pre-Arbitration* response with the merchant and hey! You'll get your money back. 

Oh, also, can a teller refuse to issue rolls of coins if they suspect the customer intends to use the coins to punch someone with, or to destroy a paint job?

I highly doubt this. Related story, you can write anything you want on the memo line of a check and a cashier cannot refuse to deposit that check. So on your next rent check, try writing "crack" in the memo line. Come on. It'll be fun. (The teller may not even read it and I highly doubt law enforcement would waste their time on it. This would get passed by.)

And one more: how illegal is it really to alter bills in some way, like drawing on them?

This one I actually don't know. I imagine it is similar to defacement or vandalism, though.