According to Dan: The nights post within a week of spending, regardless of statement close.
They expire 12-13 months from posting and can typically be extended by 1 more month by emailing Hyatt.
Best offer out there for now: (Thanks DoGo!)
Go to Hyatt.com make a reservation, click book, and before cc page you will see offer for card with the 2 free nights PLUS $50 statement credit. Looks something like this
Anniversary nights appear in account before annual fee posts.