A Crown Prince - the prince who is the designated successor to the crown - becomes king immediately upon the king's death. James would not still be the crown prince more than year after his father's passing.
Prince James bats a snowball thrown by Anna, and does it American baseball style. While a European prince would likely not know how to hold a baseball bat, the American actor playing him would.
The movie was made in the summer. You can see an air conditioner in one of the windows, in the house, in the background of the main movie photo.
After the parade, the Prince steps in a snowy, wet hole. Anna insists that they ride home in a horse drawn carriage to avoid frostbite. Once home, the Prince says he will change his socks, but everyone decides to decorate the tree. The Prince joins in without changing, so he is still in ice-cold socks and shoes.
During the parade scene, there's an antique fire truck used. The real name of the town they filmed in (Woodstock) can be seen on the truck in two places during the parade, instead of the fictional town name. (Kentsbury).