I'd arrange something with Mears for the shuttle to the port.
Labadee is a nice resort. The Malfini beach does not include food and it is a boat ride away from the main area, and it is really more of an adult thing with a limited number of guests allowed. I'd just stay at one of the many beaches at Labadee.
Dunn's River Falls is not a good idea for kids of their ages. There are many places where you have to step up high to get the next foothold, and the water is quite rapid in some places of the climb. It is a bit of a physically demanding climb for some adults, even with one of the local guides who will escort groups up the falls.
Grand Cayman and kids? Go to the turtle farm, which now is expanded with a water park.
Cozumel is quite safe. Dollars are accepted. There was some hubbub a while back about a new law requiring pesos, but it is really about banking transactions and we bought whatever we wanted this past summer with dollars. Chankanaab is nice, but so is Paradise Beach or Playa Mia. Go to Pancho's Backyard at the downtown location of Los Cinco Soles for lunch. Pancho's is in the back of this large souvenir store. Great margaritas and food. They have another Pancho's at the pier you will likely use, but it doesn't have the atmosphere of the original location.
I'd do ship's tours in Falmouth, but for Grand Cayman and Cozumel you will be fine just arranging something with a private tour - either ahead of time or at the pier. You tender ashore in Cayman, so that cuts into your time ashore. Just make sure you keep track of ship's time and not local time and you won't miss the ship's departure. Look at Shoretrips.com which is a site we recommend.