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1. Finding two Business Class SAAver award seats on the same AA TATL flights in May might prove challenging. Have you found anything yet?
2. Since AA awards are all one-ways, you do not have to return to PAR to return home: you can fly home to MSP from ATH (or perhaps from one of the Greek Isles) if you can find the needed SAAver award seats.
3. To get four nights at the same Hyatt hotel, you would book the first two free nights, and your wife would book the last two free nights.
4. Not sure if there are Fairmont hotels in any of your desired cities, but if there are, take a look at the Chase/Fairmont credit card for additional free hotel nights.