I have a mailbox set up on an Exchange server. The user wants to have access
to an additional shared mailbox. I've added it to the folder list. User can
see incoming emails and I've given him send on behalf permissions.

User has requested that when they're accessing the shared email box, that
replied items be copied to the sent items of the additional mailbox and not
the primary email box. Is this possible?

As well, is it possible to set an automatic rule for sent items in the
additional email box to be moved to a particular folder?
I've created a rule and it works for all emails but we want to only move the
sent items from the additional shared email box.

I think this would be easy to set up if we had a separate mail profile for
both accounts but the the user doesn't like the idea of having to switch
between their own and shared mailboxes. Can this be done with with a primary
mail account and an additional mailbox setup?

I hope this makes sense and thank you for your time.