If you want to check the functionality of a mailbox you can check the followings:
1.Last Login
1.1 Go to Users
1.2 Select Domain
1.3 User Control Panel
1.4. At the new page click on last logins
1.5 You can find historical data for logins
2.Change Password
At the same page you can change password of the user
3.Logging
3.1 If you want to search logs for an account you can to to Logs
3.2 Incoming Logs
You can search with various ways the log for incoming logs,
a.If you choose only domain you will get results for all incoming email for all accounts
b. The most common search is with option "To" and email address and "Subject"
The result it will be shown to Search results , you can check for example if it is spam or if the message have been delivered
4.Outgoing logs
If you want to check if outgoing message delivered to destination mailbox ,you can check outgoing logs
4.1 Choose domain
4.2 at "View Messages" click on all messages
4.3 At "sent from" write down the email address of the mailbox you want