1. Yes, we host Exchange 2010 on site for our employees. That is what these email addresses are referring to.
2. I’m assuming when you ask this, you’re talking about on our on-premise Exchange server. These people have User Mailboxes on the Exchange Server.
3. –
4. Yes, Directory Synchronization is turned on and working.
5. They do show up as @xyz.onmicrosoft.com when I look in the ADMIN panel. When I look in the directory as user, it’s a different story. See screenshots at bottom. I realize most of it's blured out but you get the idea -- when users search for these people in the address book, that's where the email address varies. You can see the top example has the correct @edu address, and the bottom example does not.
6. No, school.com can’t be seen there, but that’s never been seen there, not even when we were on Live@edu. I’m not sure why @school.edu would need to be in there when we’re not letting these people actually use Office 365 (we just need their contact info in there). Plus the fact that some of these contacts are making it over there correctly seems to indicate it doesn’t have to?