How do I export a list of everybody's picks onto an excel spreadsheet round by round? The only option I see is for the Final Four on to the championship game (by exporting a Standings Report, for example). What I need is ALL the users picks.

You can export the entire file's contents using the File -> Export menu item and then choosing the Excel option. This will get you all user information and picks so you will likely need to open it in Excel and make the necessary modifications.