We just returned from the Destiny.
As far as the towels, they are in your cabin, take them to the pool or off ship for excursions, just make sure they get back there because you don't want to be charged for them. They will clean them and give you new ones, just make sure there are2 in your room when you leave(or as many as they give you for you people in your cabin).
Shoes, Hmmm Well I would say that sneakers aren't the shoe to wear in the diningroom for dinner. I know you can get some nice inexspensive shoes at like payless. The dress for lunch and breakfast isn't dressy. No swim suits or anything like that in the diningroom but normal casual dress. You don't have to dress up for dinner if you eat on the lido deck, so your son may enjoy that better. But it doesn't hurt anyone to get dressed up ones in a while. We had 2 formal nights at dinner and 2 casual( shorts could be worn) and the others normal dinner wear.
Hope this helps.