Hi Jesper,
Before moving on, I want to confirm whether you used the domain in other mail service. If so, there might be some cache DNS records saved in FOPE. You could refer to the link below to troubleshoot the issue:
"Relay Access Denied" or "Hop Count Exceeded" non-delivery report (NDR) error message when users send emails to Office 365 users:
http://support.microsoft.com/kb/2603474
Additionally, for more information about connecting Windows PowerShell to Exchange Online, please see the information below:
Install and Configure Windows PowerShell
http://help.outlook.com/en-us/140/cc952756.aspx
Connect Windows PowerShell to the Service
http://help.outlook.com/en-us/140/cc952755.aspx
Thanks,
Linda Wang