1. DAG Active Manager Check
2. Mailbox Database Status
3. Dag mailbox database backup status
4. Mailflow test report
5. Mail queue status
If you want you can schedule it in task manager and you will have health check report emailed.
This is going to reduce some effort from the system administrators head and alert you on time to prevent a major issue.
Download link:
https://gallery.technet.microsoft.com/Exchange-2013-DAG-and-5f9dad03/file/138540/1/Exchange%202013%20DAG%20Health%20Check.ps1
Exaustive and original Microsoft Article:
https://gallery.technet.microsoft.com/Exchange-2013-DAG-and-5f9dad03
Other article
http://msexchangeguru.com/2015/06/05/e2013-health-check-script/
<-------->
You could be interested to these similar article too:
Exchange 201X- How get all users who have a forwarding email, redirect rule and delegate permission on them mailboxes or Outlook folders
Exchange 2013 - How to monitor it with powershell command
Exchange - Microsoft Exchange Server User Monitor
Exchange 2010 - Poster dell'architettura
Exchange 2010 - How to monitor Exchange Health
Exchange 2010 - How export to pst single mailbox or all mailboxes with single command or powershell command
Exchange 2013 - How to monitor it with powershell command
Exchange 2010 - How to get info on Mobile devices connected via ActiveSync, quarantine any new device and remote Wipe them.
Exchange 2003/2010 - Add a photo to user contact
