Oh how I wish I were heading to the Caribbean. Sighhhh. 8)
Anyway, let's see if I can answer some of these questions. We took our son to FL when he was 11 months. I know it's not the Caribbean, but the issues are the same.
1. We brought our stroller and our Ergo carrier along with us. Having the stroller in the airport was a huge help. Sure it's a pain with the new security situation, but we were so thankful we had it with us. You can check it at the gate right before entering the plane and they have it waiting for you when you get off the plane. Our stroller has bike tires so it made it super easy to navigate the sandy beaches.
2. My son was still nursing at 9 months so I pretty much kept that boy on my boob for the whole flight. He started off awake and then fell asleep shortly after and slept the whole way. He woke up again a little bit at landing time but kept on nursing. I'm sure it really helped with the change in air pressure. I'm not sure if yo are nursing or not, but if you are this is the way to go. Easiest flight with an 11 month old ever! People kept telling me that they didn't even realize there was a baby on the flight. 8)
3. I am an overpacker. I try not to be and have tried to learn from my mistakes. Don't bring too many clothes! You will be at the beach/pool during the day and probably won't be changing outfits too often. Some light sun dresses and a couple of nice outfits for the evening should do it. The same goes for your son. You may think that because his clothing is so small it won't take up much space, but shockingly it does.
I would bring a few small toys for him for the morning/evening time and even some of his favorite books, but don't go crazy. Being in a new environment is pretty exciting and being in the sun all day will wear him out.
4. Bedtime. Don't stray too far from your regular routine on the first night. If he has a special stuffed animal he sleeps with bring it.
You are going to want to eat out though or stay out later than his bedtime I'm sure. My husband and I fed our son at his normal time, played with him, put on pj's and then walked around our resort with him in his stroller until he fell asleep and then we took him with us to dinner. We just parked him next to us and carried on with our romantic dinner. We did this quite a few time since it was working for us.
We were also so tired from the sun that we were pooped pretty early as well.
5. You can't go wrong with digging in the sand. There is nothing a little one likes more than digging in the sand and playing in the water. When you get to the resort as the person in charge of activities what they have that might be appropriate for your family.
6. I have always sworn by Water Babies or Water Blanket or California Baby (way more expensive than the others, but works great). I have used them myself for trips to Belize and have never gotten burned. A hat is essential. I would bring two for your son in case on gets lost. We also had our son wear a SPF protected rash guard and long shorts. We still do. He's never had a sunburn.
When it comes to bug repellent I tried to use the ones without DEET, but I found that I had to sometimes. We wound up using OFF for Families. It didn't have a yucky smell and it kept the mosquitoes away.
7. Swim diapers are not only worthwhile they are mandatory at a lot of resorts and pools. Regular diapers blow up to twice their size when they get soaked. Your child will be uncomfortable and the other people in the pool won't be too thrilled either. Definitely bring swim diapers with you.
I know nothing about childcare though. Sorry.
Like I said above, we waited until our son was asleep and then took him out with us. Just try and get a table away from others if possible or sit outside. Babies sleep great in the tropical night air.
Have fun!!!! Keep it simple and enjoy your family.
Peace.