You can go to the Office 365 Admin Center, Active Users (https://admin.microsoft.com/AdminPortal/Home#/users) and you'll notice there is an unlicensed "user" for the shared inbox. If you click on this user, you can then reset the password and provide any password you want.
The answer is that the one user that was used to create the Office 365 account needed to have its login updated - it was still set to @company.onmicrosoft.com and updating it to @company.com, logging out and logging in fixed the issue. All future accounts created won't have this issue as they'll be @company.com by default.
At this time it's not a feature included in the Google Calendar web UI, by the other hand as far as I know, the Google Calendar web UI can't be extended in such way.
The alternative is to use a third party app o built your own, but that is out of the scope of this site.
Google Calendar Help
On your email client (Outlook 2013) your emails are being grouped similarly-titled emails together. To turn off this feature, look on yor VIEW ribbon in the Messages section for a tickbox entitled "Show as conversations" and untick it.
In Webmail, this feature is in a completely separate menu. Hit the dropdown arrow next to Arrange and uncheck the box "...