Yes, open any additional accounts while at the bank. Opening additional accounts will take an extra 20-30 minutes compared to 2-3 hours if one came back another day.
It does not matter. The bank accepts both and they are equally private. If absolute privacy is required, this is where a registered banking agent comes in. Through a one-time appointment of an agent with banking privileges to open a bank account on ...
One may need to try the same bank but a different branch. Most students don't have issues, but everyone has opened a trust account after a 2-3 tries. It all depends on the attorney's interpretation of the trust indenture.