Is it possible to set a specific order for all contact reports? Can they be grouped underneath certain headings making them easier to find? If so, how can I do this?
If you mean to order it on Contact Report section on the Report Page - Not yet from UI(need to write custom), they are actually ordered based on its id.
You can anyway move each report to the navigation menu(Access tab of each Report) and then order it according to your needs from
Administer -> Customize Data and Screens -> Navigation Menu.