Find email address on exchange 2010 server

 

1. Get-Recipient -Identity user@domain.com
you can use this to find user associated with a specific email address 
2. Get-Recipient -ANR user
or by using -ANR (Ambiguous Name Resolution) to find using part of a user name

Purge disconnected mailbox on exchange 2010

One option is to wait for the mailbox to be purged automatically (default 30 days)

or to use Cmdlet:

Remove-StoreMailbox -Database <mailboxDatabaseName> -Identity <DisplayName> -Mailbox