Rome is going to be a very long day & hard work on the feet, and to make the most of the time there, visitors should arrive with a good knowledge of what they are going to be looking at. This means that you may find your 7-yr old is already bored of the city before you arrive.
Pompeii is less of a chore, but again for a youngster it could become repetative very quickly.
The pyramids on the other hand could be of more interest (apart from the long coach transfer depending on where the ship docks) as it is a lot more accesible and hands on. As far as security goes, most tours have armed escorts, and the most trouble you may have is 'in your face' salesmen trying to push trinkets, or camel rides.
Something that may keep the children in high spirits is if you can arrange a couple of days in Barcelona before your return, if you were able to travel in September you can take in the Illa Fantasia waterpark (supposedly the largest in Europe) or the Portaventura theme park with the largest rolercoaster in Europe which is open until early October. Chances are that neither are going to be a match for the top US parks, but if it gives the children something else out of the trip, then it may be worth considering.
I can't comment on HAL or Carnival child club rules, but you should check if you can leave the ship if you still have a child booked in for the day. Many lines over here insist that at least one guardian remains on board in
case of problems.
Hope this gives you some food for thought,
Alan & Katrina
Last: P&O Oceana December 2016
Next: NCL Jade August 2017
Then: P&O Ventura December 2017
Golden, Grand(2), Caribbean, Royal, Sea(4+1 aborted)
Adonia(4+2), Arcadia, Artemis(2), Aurora(5), Azura(2), Britannia, Oceana(4), Oriana(12), Ventura(2), Victoria(5)
QM2(2), QE2(9), QV(7), QE, Caronia(2),
Oceania Nautica, Minerva, Minerva II, Fred.Olsen(8), NCL Jade, =X= Eclipse, HAL Ryndam, Aza.Journey, CMV Magellan