I've also noticed that when deleting contacts from the webmail (individually) the first contact is removed from Outlook straight away.

Subsequent contacts that are deleted remain visible within Outlook. After performing several Send and Recevies they are deleted too.

For reference, the Outlook version I'm testing against is below.

Thank you for reporting the issue with the contact names, it has been added to our bugs list.

As for the issue with the sync times for deleting contacts, you can see about setting up something like Fiddler to capture the traffic. This could help us determine where the delay is occurring.

It looks like the root of the issue is that Outlook for Mac doesn't include any fields like Display Name for contacts in the interface. We are working on some adjustments to how we send contacts that only have a Display Name, as you described, so that they can still be identified in Outlook.