Congratulations on your new baby!!! I cruised in October w/ my 15 mo (also DD 3 & 5). I think that it will be way easier than you think to cruise w/ a baby.
I wouldn't plan on hanging out in the room if baby is fussy, but baby will nap and probably go to bed earlier than you. Hubby and I took turns doing nap and bedtime while the other hung out around the ship. We did use the nursery after dinner and one night for Palo. It is great!! Our daughter would not sleep there though. Maybe yours will!
DCL will provide a pack n play for sleeping as well as a diaper genie. The tub is great for baby bath time.
By 12- 15 months, your baby will be able to eat table food and the Buffet has plenty of options for breakfast and lunch. At the MDR the servers brought fresh fruit and steamed veggies to our table as soon as we sat down. She then ate whatever we did- chicken, fish, pasta. We asked for a quicker pace for dinner and were out in about an hour each night except for the animators palate show night. I had a gallon zip lock back labeled "meals" that contained: disposable place mats (babies knock over/throw plates), a long sleeved bib, disposable sippy cups, wet wipes, and a few small toys. I also brought some applesauce pouches and dried fruit for snacks in port.
As far as activities go, there are "toddler times" with toys and counselors offered in the navigators, my daughter liked the "wake up with disney junior" activity, she also liked meeting the characters at breakfast, but some babies weren't loving the characters. The splash pad was fun. I brought a collapsible bucket and some small bath toys and let DD play by the pool while I sat in the lounger and watched the other 2 swim.
In port, we went to the beach, did an dolphin encounter, booked a private charter around Cayman- basically she was along for the ride. That's our style at home too, though, we bring our babies everywhere.
We ended up not bringing a carseat... There were a couple times in port that I wished we had. The car service that took us to and from port had car seats and I didn't need them at WDW. It's not illegal to hold the baby in a cab and the other excursions we took were on big buses, but I was not totally comfortable.
We had a stroller for WDW- a Maclaren that folds up small and fit easily under the bed. We did not use it at all on the cruise.
I brought a few toys and books from home, but they did not get played with at all and were a waste of space. I bought sand toys at the first port and she played with those and other souvenirs that we bought.
Hope this helps.