If you want to be absolutely sure the kids have a GOOD onboard program where you can confidently leave them while you go on certain tours Carnival is a better bet. The kids will make friends and be happy.
Now - on the other hand, sometimes kids on vacation who make new friends do not want to leave them and go touring. With HAL you are more likely to find yourselves tending to your own kids most of the time. HAL jas kids programs, but if it is a smal shiup therte may not be many onboard and that makes the programs less fun for the kids.
As far as seeing Egypt, the pyramids are perfectly safe if you stay with your tour - just don't wander off. And this is not a joke - don't try a camel ride. The trick is to have the camel "take off" and a stanger come to the rescue, who will then ask you for money.
Rome is a lot of walking, unless you hire a guide in the city, or take an organized tour. If the cruise starts there arrive two days early and see as much as you can. It is definitely worth a visit. Athen only requires a day to see the acropolis. Ephesus is a great visit. Santorini is beautiful, same with Florence and your kids will love Pisa.