Some thoughts. Yes, you can wear sneakers. We travel extensively in Europe and in particular a lot in Italy. You have to have very comfortable walking shoes or sneakers. You cannot/should not wear shorts in the Vatican. A good practice is to take a real good back pack which you can also attach to a suitcase as you walk. With the right backpack you can pack a lot of stuff.
There was an old rule..Pack your suitcase and then take 1/2 out. Also the backpack can go under the seat. PUT ALL VALUABLE THINGS IN THE BACKPACK AND PUT LOCKS ON THE PACK. DON'T LET IT OUT OF YOUR VIEW.
When you leave the ship or if using an apartment/villa you can always take the backpack and not leave key stuff behind.
If your driving in Europe don't leave valuable items in view in the car. They don't have the violent crime (usually) but they do have pickpocket, thefts, etc. Regarding dressing up on the ship. It is optional and you have to ask yourself if it is worth carrying all of that stuff for a couple of nights walking around the ship? Actually, it is fun sitting in the lounge and watching the ones that dress up.
