To your points above:
#1: Yes, all nights will count towards 2021 status
if you check-out Jan 1-4 of 2021. You can read the details of the promotion at this link:
World of Hyatt Bonus Journeys | 3x Points. The FAQ clearly states how the tier-qualifying nights will count towards 2021 status, see FAQ #23..."If your stay starts during the Promotion Period in 2020 and your checkout date is January 1-4, 2021, your Tier-Qualifying Nights and bonus Tier-Qualifying Nights will all be awarded in 2021." Once you achieve Globalist status in 2021, your benefits are good for all of 2022 and expire in Feb 2023!
#2: Just use the Hyatt website to search for prices using their map search feature. Enter a location, like Chicago, select your dates, and then scroll the list of properties that come up listing the prices. You can move about the map (check the box for Search as Map Moves) and the prices and locations will update as you move. Depending on the dates you select you will get different price options. I selected Dec 11 - Jan 2 and found the following:
3 HP in Chicago ranging from $59-$62
HP Los Cabos $65
HP in North Charleston, SC $54
Excalibur in Las Vegas $34 (not sure about resort fees here and they jack up the rate for New Year's Eve)
Try this yourself using date ranges that fit your schedule, just make sure you have a check-out date set for Jan 1-4 range of 2021
#3: I'm not sure about the Mexican option and 3rd night free.
Remember, you don't have to try and get Globalist status all in one lengthy stay in December with an early Jan 2021 check-out. If you are planning to travel more in 2021 then this promotion will give you a good boost to starting out 2021 strong. Even if you just stayed 15 nights, Dec 17th - Jan 1st for example, you would get 30 nights credit towards 2021 status and you are half-way to Glob.