Functionally, a deputy appointment acts like a form of limited power of attorney. However, I've not seen anything that suggests that our deputy appointment forms are legally recognized as powers of attorney. SO we're not including the notice or acknowledgement on our deputy appointments, and we have not been advised to.
That said, it's a good question, and it's probably worth checking with your bank's own counsel.
To your second question, a deputy appointment applies only to a specific lease. So even if the appointment allows the deputy to surrender the box (some do, some don't), it could not be construed as allowing the deputy to enter into a new lease. SO the "exchange" to a new box simply won't work.