I haven't cruised with my 14 month year old but we have traveled quite a bit (flew/drove/etc) We are cruising with him in September so I will better be able to answer the cruise specific questions in a few months

Some of your other questions I can give you some advice as we babywear and cloth diaper.
I would be surprised if your LO still fit in the moby. Don't get me wrong I LOVE LOVE LOVED my Moby when he was tiny and I wore him all through many many airports in it, but by 1 year my babe had outgrown it. With that said if you like a Moby wrap then you might want to consider a woven wrap...same concept but more support and more options (HELLO BACK CARRIES!) My curious 1 year old loves to be able to look around. We plan on bringing our wrap and a umbrella like stroller (summer infant with a shade...it was like $40 but super easy to use) My babe isn't really happy in the stroller unless we are moving, so I might even leave it because he will not just sit in it and if we are walking to and from a destination. I would rather wrap him. I also want to invest in a water sling for CC.
On the CD. I would suck it up and try disposables. We tried to travel with cloth and we found it was difficult to manage. Transporting the dirty diapers while traveling (even by car) was gross and difficult. Also I know that pods and other simple laundry devices do not bode well for cloth. So packing detergent was a hassle. Maybe I am just too particular on my detergent. Plus I know that some public laundromats won't allow you to wash diapers and I don't know about a cruise specifically but they are such germaphobes they might not allow it. Also if you are on the beach all day do you want to have to haul wet and poopy diapers back onto the ship? We use Honest diapers in such situations and I love them as they have zero odor and they don't have leak issues. I bought some nautical ones just for our cruise in Sept.
For the car seat...we never fly with ours (that is our choice I know many people say that the proper thing to do is buy our infant a seat and have him in the seat...no flaming I KNOW THAT IS WHAT IS RECOMMENDED BY THE CARSEAT NAZI'S) there are a lot of baby rental places that you could "rent" a seat from for your LO. They could deliver the seat directly to your friends house and then pick it up from their house as well. This is what we have done when we have flown and needed a seat once we arrived at our destination. It is a couple of bucks a day and the expense is worth it to me to not have to deal with the seat on the plane or the boat. Honestly it isn't even that expensive.
I hope that helps.