Can iPhones with ActiveSync use Exchange shared mailboxes?
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.