How many CC's do you have with Chase? If only the Hyatt Visa and the Freedom, I have two suggestions for you:
1. Chase Sapphire Preferred: BEFORE YOU CANCEL THE HYATT CARD, Apply for Chase Sapphire Preferred online for 50k Points after 4k spend. That will get you about 55k points right there. Add your wife as an AU and get another 5k points - for a total of 60k+ points. Transfer these to Hyatt and you have enough points for 2x nights in Cancun + 10k points to spare. I Say to apply before you cancel because if Chase denies you or asks you to call in, you can offer to close your Hyatt Visa or shift credit over to the Preferred card.
For an extra bonus, have your wife apply for the same card via a referral link that you will send from your account. If done correctly you could net:
Chase Sapphire Preferred:
50k Bonus - You
50k Bonus - Wife
5k Points - Your initial Spend
5k Points - Her initial Spend
5k Points - You adding her as an AU
5k Points - Her adding you as an AU
10k Points - You sending her a referral link
Total of 130,000 Chase Points - Enough for 5 Nights at Hyatt Zilara Cancun @ 25k points per Night.
2. Chase Sapphire Reserve: If you are near a Chase Branch, go in and have both you and your wife apply for the card with the 100k bonus each. There are currently no referral or AU bonuses with the card, but between the 200k points you guys will net with the bonus + the 10k points you will net from the spend, you will have enough for 8 nights in Cancun on the points alone.
As far as the Hyatt Visa goes, if Chase doesn't require you to cancel it to apply for either of the cards mentioned above, then move the credit line over to your Freedom and cancel it anyway. In 18 months or so months you can reapply and get the card(s) again for the Hyatt free nights.
You could then try to repeat this process next year with whatever card you don't end up applying for this year. Essentially rotating the Reserve, Preferred, and Hyatt Visa every 24 months on a staggered routine to maximize the bonuses (Although at some point you have to assume Chase will probably stop this at some point)
As far as your other questions go:
1. Yes you get to keep your Hyatt Points earned if you cancel the card
2. Probably will lose Platinum status but with Hyatt IT you never know
3. No benefit other than category 1-4 free night for when you cancel
With regards to your credit, closing a card only hurts if it negatively impacts your age of accounts, credit to debt ratio or if it was closed by the bank for lack of payment. By doing what I suggested above, you and your wife will probably take a 3-5 point hit temporarily, which will quickly bounce back after a month or two if you continue to keep your accounts in good standing.