Ahoy, James!
On any Disney Cruise, there is be a
Pirate Night, where your fellow passengers will be dressed in their pirate-y best outfits. On these nights, I have also seen many little girls in their cute Tinkerbell dresses.
All cruises on the Disney Dream also include an optional "dress up" night, where some cruisers choose to dress up while many others keep it cruise casual. This would be an ideal night for your daughter to dress up as her favorite Disney Princess.
Also be sure to check at Guest Services to see if there will be a "Princess Party" on your sailing. This is a ticketed event (but free) where you might have the chance to meet up to five different Disney Princess all in a row, only standing in the first line. Another great Princess dress up opportunity.
As far as shorts at dinner, they are technically allowed. The dining rooms are usually chilly, so shorts might not be ideal. The official dress code simply defines "
cruise casual" as no swim suits or tank tops. It is your vacation, so dress how you wish.
Happy Cruising, Tracy V.