I'm Lucy, and I'd be happy to answer your questions today.
1. Yes. However, the other party is required to serve you with a copy of the subpoena before sending it to the bank. Fed. R. Civ. P. 45.
2. Yes. But again, the other party has to alert you - it's just not the bank's job to notify you.
3. They must send whatever is requested by the subpoena.
4. They do not need consent. They cannot refuse to send the documents. A subpoena is a court order, and it must be followed.
5. To stop them from subpoenaing the information? Not really, other than voluntarily providing all the information yourself (so you have control over what they get) or settling the case before they start discovery. If you want to stop them from getting information requested by a subpoena, you can file a Motion to Quash, as provided in Rule 45(d)(3). If you're planning to file the motion, you can contact the bank's legal department, send them a copy, and ask them to wait to send the documents until you get a ruling on that motion.
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