Hi Brandee;
Best to get connecting cabins which will also give feeling of more space. If you go for balcony cabins I believe some lines can even open the balcony partitions to make on big balcony. A bigger cabin depends if you are talking Penthouse type with a separate bedroom for kids and sitting area as well as 2 bathrooms (NCL). A single room will be crowded for 4 and limit privacy in a major way especially for the situation you describe.
Have sailed Carnival a few times and they are fine. 18yr.old will want to doing her thing and kids program is good for 14 yr old. On all the lines we have sailed on the kids programs have always been well run and my now 12 year old has always enjoyed them. Kids are grouped by ages so your 14 yr old would be with 13 to 16 group. Have you thought about inviting a friend of the 18yr old? You, your husband and your 14 yr old could share one cabin and they the other? I think she would would enjoy being with one of her peers for the fun factor and really make it special for her and you since you wouldn't be seeing them that much. Of course the 14 yr old might feel left out and would hang out with you if he doesn't connect with kids in program.
Our experience is that Carnival, NCL, Royal Caribbean and Cunard are all very family friendly and provide plenty of activities for kids and adults.
As long as kids are well behaved there should be no problem with them roaming the ship. As always they should be aware that they are among strangers and should use common sense.
As far as what my children have liked best is the kids programs and interesting shore excursions. Those would depend on their interest (ie beach, shopping, exploring etc) as there are all sorts of options.
Better check on passport option for younger. I don't know details for kids, but for adults you need to have . Our 12 yr old has a passport.
Hope this helps;
Sam