In Jamaica I would prefer to book a tour with the ship... we have done Dunn's River Falls (crowded, but awesome experience, however Falmouth is farther away for that one I think) and a river tubing tour in the past.. both were great, however, I'm not sure I would do either with a 6 year old... maybe someone else can chime in.
Grand Cayman -
DCL pulls to the main port in 'downtown' / the capital - it is super easy and safe to get pay for a spot mini-van to 7 mile beach (various stops, we've done a couple...). We've also done Stingray City, which was fantastic and we booked it with an independent tour that left right from that main port. I mention the "main port" because other cruise lines pull into a different port in the far south... I recommend a booked tour if the ship is diverted to that port because it takes 45 minutes alone to reach the "downtown" one way, and the cab/minivan ride is prohibitively expensive from that pier/port. That said, DCL has always tendered us to the "main port" … it's just something to keep in mind.
Cozumel... it's very easy and safe to pay a cab (set prices) to various beaches... we also did beach days with DCL.... I will be very honest... I was not that impressed with any of the 3 beaches (sand wise...) or food wise. However, we once did an excursion with a jet boat to a beach to the far north that is on a separate island and that was spectacular... but because the boat is very fast with super fast turns I believe only older children are permitted - that excursion is fantastic and the beach is sugar white and like a dream and food was very good... We also did an excursion to the mainland with the catamaran and then to the pyramids... we did this on our own, but I highly recommend booking a tour for that one - even though we did make it back to the ship on time.
We also did the dolphin experience in Cozumel with DCL and it was really wonderful.
I'm happy to answer more questions - I might still remember the names of the excursions and beaches …. but I'm currently blanking on it.
P.S. regarding: Snorkeling with kids - we started doing it with our oldest who was 5 - however we snorkeled from the beach (walking in on
Castaway Cay and from 7 mile Beach, Grand Cayman)… we had brought our own snorkel and mask to make sure it's a comfy fit. I'm not sure if I would snorkel in deeper water off of a boat at that age, probably not. Also of course, kid should be an excellent swimmer...