Is there any way to view the disabled mailboxes (closed mailboxes) under Active Directory (AD) users and computers? Also, can the Exchange administrator receive an e-mail notification for the disabled (closed) mailboxes?
We should first review how this works. Each typical user of an Exchange environment has an account that they use to log onto the network with (i.e., an account in Active Directory) and a mailbox within Exchange that is associated with that account. There are different ways to decommission a mailbox once a user has left the organization. I'll discuss two of these in answering your question.
Option 1: Hard delete after x days –- in this approach, when a user leaves an organization, their account is typically disabled, blocking them from logging onto the network. The mailbox continues to be associated with this account, and can be logged onto using alternate credentials if someone else needs access to the mailbox. A mailbox with a disabled account will no longer receive messages. Typically in this scenario, the mailbox and associated account are completely deleted after x days. In many cases, decommissioned accounts are moved to a dedicated Organizational Unit (OU) reserved for this purpose.
Option 2: Re-assigning SMTP Address -– in this approach, the departed user's account is disabled, as above, but their SMTP address is removed from their original mailbox and moved to another object such as a "catch-all" mailbox designated for all terminated employees, a manager's mailbox or similar.
So now to answer your questions. Viewing a list of all disabled mailboxes really depends on how you've decommissioned them. If you simply want a list of all accounts that have been disabled, these are visible within Active Directory Users and Computers as users with a red x through them. If you're using Option 1 and have moved decommissioned accounts to a dedicated OU, then this is even simpler. Just point Active Directory Users and Computers at the OU and you'll have your list. If you're using Option 2, then you could use View | Show Columns within Active Directory Users and Computers to add the "E-Mail Address" column. Sorting by users without an e-mail address will allow you to identify those who have been terminated.
Finally, you asked about an administrator receiving e-mails for the disabled mailboxes. I'm assuming you're asking whether administrators can receive messages designated for the terminated users, in which case Option 2 provides a solution.