Your first international segment is SYD-AKL. As you are starting your trip in Oz, you do not have the 7 day restriction. You can make reservations and changes anytime. You must be ticketed not later than 1 hour before your first flight. Of course, if you attempt to make last minute reservations on any of your segments you might find that the class you are trying to fly is not available.

You left off a critical segment. Your last segment must be LAX-SYD as you need to complete the RTW and end in Oz.

The designation for the international airport in Buenos Aires is EZE.