I did this itinerary back in October, only we docked in Ocho Rios, Jamaica, and not Falmouth. I would highly recommend a ship tour if you did Dunns River, or any tour in Jamaica for that matter. If not a ship tour, then maybe look at Shoretrips.com to see what excursions are offered there. I definitely wouldn't wander off on your own in Jamaica. As stated, merchants can hound you at Dunn's River. Water shoes are a must for Dunn's River, and they do sell them onboard the shops on the Freedom if I remember correctly. Dunn's River is supposedly the "must see / must do" type attraction if you're in that area.
I can't say much about Passion Island in Cozumel. When I was there, I just went to Paradise Beach my last time there, and Mr. Sancho's Beach Club the time before that.
A taxi to Seven Mile Beach should be fine in Grand Cayman. Be advised, this is a tender port, and not a docking port because of the shallow waters close to shore. Because of this, people with excursions booked through the ship will get priority in boarding the tenders.
Also, not to dampen your spirits, but there may be a chance if the seas are rough, that the tenders may not run in Grand Cayman. This happened to me the last time, and the port fees for that port just got refunded by means of an on board credit, and the excursion I had booked also got refunded. For me personally, it was going to take a lot more than a missed port due to weather to ruin my cruise vacation! Hopefully that won't happen to you.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)