Can a bank make a customer liable for the first $50 across the board in order to recover their costs of "charge backs"? Which essentialy leads them to refuse to investigate errors that are less than $50?

No. You can only impose liability for unauthorized EFTs in accordance with Section 205.6. To impose liability you have to make a reasonable determination that the transactions in question are, in fact, unauthorized. And if dealing with certain signature-authorized debit card purchase transactions, you won't be able to impose liability under the MasterCard or Visa zero liability protection rules.

You may decide that any claim of $XX or less will automatically be paid, without investigation (make sure you have controls in place to prevent repeat claims and obvious scams, or you'll get eaten alive!), but you cannot take away protections found in the regulation.

If you disclosed a fee for claims under Reg E, you can do that. But the fee you charge can't have a "chilling" effect on the customer and you can't charge it on valid claims.

Is it permissible for a bank put the responsibility on the customer to go to the vendor and try to resolve the dispute themselves or since the customer came to the bank, is it the banks responsibility?

It's the bank's responsibility to resolve with the vendor. All the customer has to do is report it to you. You're the card issuer therefore, it's your problem.

If it's a Reg E dispute the customer is not requried to go to the vendor.