Can iPhones with ActiveSync use Exchange shared mailboxes? I have a team that all have privileges configured in Exchange to give them read/write access to the Inbox, and send-as privileges, for a shared mailbox.  This works great on Outlook, where they can log in with their credentials and just pick "Open these additional mailboxes" in Outlook's account settings.

There doesn't seem to be anything precisely parallel to this in the iPhone, and the Googling I've done suggests that the easy way won't work on the iPhone.

But where things get really odd is that I don't seem to be able to add another Exchange mailbox on the iPhone with the shared mailbox's username and password.  Every time I try to authenticate, I get a password error, despite validating the server settings and the password repeatedly.

How can I get this team the ability to receive, move/delete, and respond to emails for this shared Exchange account from their iPhones?
As of August 2013, Exchange ActiveSync doesn't support shared Exchange accounts.  This isn't an iPhone issue, but rather with the ActiveSync protocol.

One option would be to use Outlook Web Access to access the shared Exchange account.  This would allow the team to do everything that you've outlined here.  

Another option would be to set the individual Exchange accounts for the team to automatically copy mails from the shared account into a folder on the individual's Exchange account.  However, I don't think that this will meet your needs, especially moving/deleting and responding to mails from the shared account.  I mention it for the sake of completeness.