Welcome to Hyatt!

I have copied the relevant portion of the terms and conditions for FFN below.

To qualify, you must be a Hyatt Gold Passport member, provide your Hyatt Gold Passport account number at reservation/check-in, be a registered guest physically occupying the room to which you are registered, pay an Eligible rate, and make full payment on a valid MasterCard card for each separate stay. "Eligible Rate" is any hotel published room rate, including, but not limited to rates found on hyatt.com, hyattplace.com, hyattsummerfiedsuites.com, prevailing rack rate, BAR, volume, AAA and senior citizen rates, etc. "Ineligible Rates" are discounted rates, including, but not limited to, any free night stays, Internet wholesale rates (such as priceline.com, hotels.com, etc.), traditional wholesale rates (such as GOGO Worldwide Vacations, Pleasant Holidays, etc.), airline crew rates, airline employee rates, travel agency employee rates, Club at the Hyatt discount certificate stays, Hyatt Vacation Club stays, Hyatt employee or employee family discount rates, airline interrupted-trip vouchers or contracted rooms (a contract room is a room that has been reserved pursuant to a written and executed agreement between a hotel and a corporation, government agency or individual for a negotiated room rate in exchange for an agreed upon number of rooms to be rented for an extended period of time).

To answer your specific questions, yes, AAA rate does count toward FFN. As shown in the T&Cs, if it can be booked online through Hyatt it will count. A 1000 GP point bonus is one option for the diamond amenity. Hotels also offer various food/beverage options such as a cheese or fruit plate with a bottle of wine (often a half bottle).