Stay 4, Pay 3 - Hyatt is indeed only giving 3 elite qualifying nights.

I am not aware of any issues this year with posting. I second what others say: most points post the day after, the rest on the second day after check-out.

No need to wait for any printouts etc. Just ask them to mail you the folio.

And no need to "mentally track" flights and hotel stays at a daily basis. Go through it at the end of the month and you'll be fine.

Also: instead of contacting Hyatt customer service, you can get in touch with the hotel and let them check. Most likely a better and easier experience.