Reasons for the war?

War is hell.

SPOILERS! It seems that Madame Sullivan started the war to bring all the rouge wizards and witches in line. It was probably she who turned the neighboring country's prince into a scarecrow. His absence started the war. Because of the war, Sullivan got to call all wizards and witches to the castle to join in the war effort, so they wouldn't suspect a trap. But for some it was a trap, like the Witch of the Waste, who had her powers taken from her. And it was a trap for Howl too, but Sophie saved him. At the end, when the dog revealed to Sullivan that the scarecrow was restored to his true form and would go back home, she realized her scheme was over, and called off the war.

The Prince was kidnapped