Welcome to cruise-chat! Dr. Livingston has done a complete job of answering your questions. There isn't much more to add, other than there is also Sorrento's Pizza and the Cafe Promenade. The cafe has extra-fee coffees and some extra-fee desserts, but it also has great little sandwiches, cookies, and pastries, which do not cost you extra. The cafe is open 24 hours. Sorrento's is open most of the time. Pizza and other Italian snacks are available - included in your cruise fare. While there are three separate names for the levels of the dining room, it is really one big dining room and you are assigned to a specific table on a specific level for dinner. They usually only open one level for open seating breakfast and lunch...this seems to be the lowest level most often. Dining room breakfast is always a great idea since it isn't as hectic as the buffet.
And yes, it will only take a few minutes to get from FLL to the port. The airport is essentially across the street from the port, but then it isn't a straight shot either.