What is Azureedge net used for?
Azure CDN endpoint is a subdomain of azureedge.net, which is used to delivery files using HTTP and HTTPS. To create new Azure CDN endpoint, navigate to All resources |CDN profile | +Endpoint.
What is Outlookdiagnostic Azureedge?
The Microsoft Outlook team uses the outlookdiagnostics.azureedge.net hostname for delivery of their support and recovery assistant - e.g. from https://diagnostics.outlook.com/. For additional information on the specific application I would recommend posting in the outlook MSDN forum.