You're pretty much out of time. You should really call Carnival and get the correct answers.
While your American friends have many more answers than I (from Canada) can offer you, but I will say this. It has been my experience that there are ports in other countries that you MUST supply a passport to enter they country. Now unless your cruise is a cruise to nowhere, it is possible that your children may not be able to get off the ship and visit SOME ports.
Again, you really need to call Carnival and get the low down.
Western Caribbean - Elation - April, 2015
Southern Caribbean - Triumph - October, 2015
Exotic Western Caribbean - Dream (B2B) - November, 2013 (Our Honeymoon)
Eastern Caribbean - Dream (B2B) - November, 2013 (Our Wedding)
Mexician Riviera - Spirit - November, 2011
Western Caribbean - Freedom - January, 2011
Southern Caribbean - Miracle - November, 2010
Western Caribbean - Liberty - November, 2008
Eastern Caribbean - Valor - October, 2006