I´ll answer your questions in reverse
1. Citi Prestige can´t book it. This isn´t compatible with Citi Prestige / free night promos. This must be booked via a Prive agent on a flex rate/BAR.
2. Prive rates are equal to BAR - Best Available Rate - which I believe in Hyatt speak is the ´Hyatt Daily Rate´ (not a Hyatt junkie so someone tell me if I´m wrong here as to the terminology). BAR = the lowest FLEX rate (we´re not talking about advanced purchase rates here).
3. I´m not sure if this list is open to the general public, as I´m logged in as a Prive travel agent, but you can try this link:
https://www.hyatt.com/prive/exploreHotels
There are 146 participating hotels. I believ ALL Park Hyatts participate, most Andaz participate, and a good amount of Hyatt Regency, Grand Hyatt, and Unbound participate.
For those who are familiar, this is Hyatt´s version of Four Seasons Preferred Partner, Mandarin Oriental Fan Club, etc.
And yes, this is a great way to NOT chase 60nts, not obsess over your upgrade clearing, etc etc.