If you want to share contacts with a role click here to send a request to the Help Desk.
File --> Open --> Other User's Folder each time they want to access it. To avoid that and have the mailbox show up in your list of mailboxes please click here to send a request to the Help Desk.
If you continue to get calendar notifications even after you've moved departments check with the originator of the appointment to see if they still have you set as a delegate.