Yes, the dress codes do apply for kids, basically in the dining room for dinner, thats fine for breakfast and lunch. Packing for Alaska is tricky, but pack plenty of clothes that you can layer and peel off as needed during the day. I did pack one pair of shorts and did wear them, but I wouldn't count on that. Also rain gear will most likely be needed, you can expect to run into that at some point.
You can find a lot of good tips reading through this Alska guide for 2003: