hi martha
like you said it is a crazy itinerary, too many places to see in that short of time, plus the kids will not enjoy moving constantly, (getting up packing checking out transportation get to station or airport, security and passport check, where you going how long checking into to new place) it's a long day. plus i don't think car is useful, parking near or in a parking garage further away, cost of parking and not using car, public transportation or a taxi. Fly from barcelona to genoa, spend couple days before cruise. walk around the town, an aquarium, looking for lunch place with balcony and sea view, gelato. check out timeout.com genoa, (a train going up hill for kids) apathtolunch.com for area info, seaside access to put feet in water. board cruise and when you come back decide what's next. look at times flying out, too early to check in to hotel and early checkout to catch flight, fares to which city you want before going back to barcelona. check booking.com and cross-pollinate.com, put in filters (A/C, washer/dryer, lots of stairs to climb or elevator (lift), deposit, cash only read fine print). others on this forum will give you excellent pros and cons about your travels, they are here to help you out, and keep you sane and not stressed. happy travels
aloha