I think a Med cruise with kids of any age is a good cruise! It is a good way to see a lot of different things and places, and as long as you don't try to "see everything," you will have a great time.
On our first Med cruise in 2011, our kids were 11 and 6 at the time. I just looked at the itinerary for your cruise, and can talk a little bit about the ports and what we did.
Barcelona We arrived three days early and had some lazy days adjusting to European time. Barcelona is a beautiful city with a great subway system that will get you anywhere you want to go. There are plenty of parks where your kids can run around, and lots of interesting things for them to see. Spanish culture is very child-friendly, and there will be kids out at all hours of the evening. I would suggest getting on "Barcelona Time," which means eating late (around 9-10), and napping or relaxing in the early afternoons. As far as places to stay go, I would recommend "apart-hotels," like Erik Vokel Suites
http://www.ericvokel.com/hotel?lg=en&id=barcelona-bcn-suites or Hispanos 7 Suites
http://www.hispanos7suiza.com/en/, where you can spread out and have a little bit of privacy. We've stayed in both apart-hotels and liked both of them. They are more convenient than a hotel room, especially if two kids need to crash but the rest of you feel like it's 10am! (Or you need to crash and the kids are raring to go, which is more like our family!) We loved Barcelona. It's a great way to start the vacation.
Naples is an easy port to do with the kids, because you can walk right into the city from the boat. Or, if you are going to head to Pompeii or anywhere on the Amalfi coast, there is a taxi stand right at the port. Or, take a kids-themed Disney port adventure. I recommend Pompeii, though that should really depend on your kids' interest levels. Not every kid likes to walk through ancient ruins.
Civitavecchia and Livorno are both ports that require some transportation to the featured cities. Expect to spend about an hour getting into Rome and an hour and a half getting into Florence, both by car. If you want to visit those cities, expect to be up and off the ship pretty early, to allow for enough time to get there and back. We used Disney transportation to get into Florence, and we visited the city center and museum on our own. For Rome, we hired a private company and driver because we wanted to see a lot. Unfortunately, there is not a whole lot of time in either of these ports, so my advice would be to pick one big thing to see (good kids sites are the Leaning Tower of Pisa, on the way to Florence, and the Colosseum in Rome), and spend the rest of the time walking around, finding fountains, old windy roads, and gelato shops on your own, allowing plenty of time for relaxing under umbrellas in cafes. I recommend Rick Steves Mediterranean Cruise Ports book, if you want to do some of your own planning. Some of the Disney port adventures are great, and others are like forced marches. I'm a fan of getting into a city and going off on our own. Following 75 other people on your bus is a total drag, in my opinion, and doesn't allow for parents to adjust their plan in relation to their children's energy level.
Villefranche is a beautiful, tiny port with quaint streets, cafes along the water, and a rocky beach that is perfect for swimming. From Villefranche, it is easy to get to Monaco, Cannes or Nice from the train station, which is within walking distance from the port (this is a tender port, so you will take a boat from the Magic to the port). My recommendation would be to use this as a beach day, since you will likely be worn out from the previous port days. You can take the train to Cannes, which is lovely and large, or you can just stay put and use the rocky beach at Villefranche. The restaurants in town are convenient and serve delicious food. It's smaller and quieter than the surrounding, more exciting, cities, but that's what I like about it. August in the Mediterranean is hot and crowded, so all our choices are made with this in mind.
The Med cruises with kids are a lot of fun, and are a great way to see a lot of things in a short amount of time. On reflection, the best piece of advice I can give you is to temper your expectations, and follow your children's leads. Probably you should not expect to see the Colosseum, Forum, Vatican, Sistine Chapel and Trevi Fountain all in one day. Probably your kids are not going to be interested in seeing Renaissance paintings at the Uffizi, either. Choose a few things that interest the kids, and go with the flow. It will be very hot and crowded, so allow for frequent rests (and find places with bathrooms, since they seem to be few and far between just about everywhere in Europe!) and diversions from the expected path of the day. You don't have to get off at every port, either. As far as dinner seating goes, we prefer second seating on the European cruises (I think it's 8:30 or 8:45). It's late, but we were always back to the ship late. But main dining is fine, too. If you miss it, there's always room service or quick service food. Or you can just show up a little late and they'll get you fed and out of there. One last thing I wanted to mention was the availability of the kids clubs during port adventures. On our first trip, the kids stayed on the boat while we took a few hours to visit Pompeii on our own. It was a good way to see the ruins on a hot day, without dragging bored kids behind us. They stayed on the ship, had fun, and ate lunch, and then we joined them in the afternoon. I don't know how your family feels about something like that, but I just wanted to throw that out there.
PM me if you have any specific questions or if I haven't been clear on some points. You will have a great time! Go!