Hi Mike,
Based on your description, I did some tests on my Outlook for Mac 15.6(OS X 10.9.5). When I opened an .ics or .olm file, Outlook for Mac created a new message and attached the file. That is an expected behavior because Outlook for Mac cannot load .ics file. However, I was able to create new appointments and meeting invitations on my Office 365 calendar. The test in Outlook Web App also works well.
In this situation, you can try some additional steps for narrow down.
1.Open the .ics file and log in to Outlook Web App to see if you can create meetings in your calendar.
2.Reinstall your Outlook for Mac.
3.Connect your Office 365 account to Outlook for Mac on another Mac to see if the issue is related to your Mac.
If you can confirm the issue is related your local client, it is recommended that you contact our specialist Outlook for Mac Forum for effective assistance, since our forum mainly focuses on Office 365 Online services.
Please note that the information above is suitable for an Office 365 tenant that is bought from Microsoft directly. If the affected Office 365 tenant is bought from a Microsoft syndicate partner , such as Godaddy or AppRiver(you can contact your admin about this), I suggest contacting your partner support directly for dedicated assistance since the structure and some management interfaces are different. Your time and understanding are highly appreciated.
Best Regards,
Alex