L.S.
Oh fun! Traveling with kiddos.
You can take the stroller with you inside the terminal. You would check it at the gate when you board the plane. Same goes for the car seat unless your child is going to sit in it during the flight (must be a 5 point harness type approved by the FAA). Just see the person at the gate before boarding for help.
Re the beach... We went to Cancun when my ds was about 18 months. Take a sunscreen, sun blocking shirts, hat, etc. One thing that happened to us is they lost our luggage and we didn't get it until the next day. My advice would be to pack about 2 days worth of diapers in your carry on. We barely had enough to make it. Also, pack lots os snacks and food in case of delays. Snacks keep little ones busy too. We had a couple of cans of those toddler meals and cereal. My son normally didnd't eat canned food, but he seemed to enjoy it.
BTW, I'm very sorry to hear about your son. I just noticed that and my heart goes out to you.