Do we have to open a bank account in the same state as my LLC?
The answer to both of your questions is, no.
You do not need to open a bank account in a Wyoming branch of a bank. Many national banks will open a business checking for a WY LLC in any other location. For example, our company is a WY LLC and we opened our account in Chase bank in New York. Depending on a bank, they may ask you for a letter from your CPA stating that you do not intend to do business in the state where the branch is located. If you do intend to do business in that state, you will need to register your LLC as a foreign entity.
As for the address, you will need a Registered Agent in Wyoming, but that's not the same as having a mailing address in the state. That is just an agent who will receive official (legal and government) notices on behalf of your company. Your mailing address can be in any other state and can be listed as such right on the Articles of Organization.