Based on that flight time, I'm betting it's probably AA to Miami. 2 hours should definitely be enough time for that, especially since you're not checking bags. You should be able to check in at the kiosk and not go to the AA line at all. Other than AA, I'm not aware of any planes going out that early, other than small planes. If it's a small plane, I don't think you can check in at the kiosk, but you could go to the counter and check in without any problem within
two hours, for a small plane.

Regarding your hotel, when do you anticipate arriving at the hotel? That might make a difference, as to whether you would want to rent a car and do some touring around and then turn in the car the next morning, so just get a cab and go to the hotel.

Regarding your last night, assuming without knowing that you are not going to rent a car, I would agree with JD, either 1) Mary's Boon--lovely old Caribbean style hotel, on a beautiful beach--you can either have dinner at the hotel, hit and miss food there at the hotel, or a better bet, walk next door to Karakter and get a nice dinner. You won't need a car for that.
2) Someplace in Maho, probably Royal Islander or the Sonesta, if you can get them on a hotel site, as they are both timeshares. Lots of restaurants there to walk to, casino if you want, very close to the airport. Possibly Alegria. Alegria has had some issues lately, but for one night, it would be a nice location, to walk over to Maho and check it out, also very close to the airport.

Let us know if there are any other questions.