In general, the Wakatsuki family (especially dad) argues with Woody about his decisions, but they end up proudly supporting him.
Remember that the majority of the Wakatsuki family is very lucky to be together in the "camp," and Woody takes on the role of caretaker and father figure by giving the other children work fixing their living quarters. Through Woody's inginuity, they cover holes in the floor with the lids from cans and they use any trash they can to cover any other holes in the walls to keep out the harsh weather elements.
This obvious care and concern and love for his family serves Woody well later. It isn't long before some of the people with Japanese heritage are allowed to join the draft. Woody's dad argues with him relentlessly, feeling that fighting on the side of the Americans is supporting the conditions they were placed in during the war and is dismissing Woody's Japanese heritage, but Woody is adamant, and the family is eventually proud as Woody marches off to join the fight.
Another interesting element that can be related to your question is that Woody decides to visit Japan apart from his military station and visit family and friends of his father. Woody understands how many Japanese must see him as a traitor, but he finds welcome in his native homeland:
There are so many relatives to meet. Everyone will want to see Ko's son.
Meanwhile, however, Jeanne is dealing with the prospect of high school in America and the issues that accompany that in the postwar era.
In regards to the last part of your question, no, I was not surprised that the Wakatsukis were supportive of their son, Woody. In my opinion the entire family is an example of honor, pride, and perseverance in a hostile wartime and post-wartime environment. It should not be surprising to any reader to find father and mother, sisters and brothers, supporting their son. Woody believed the Americans to be on the "right" and "honorable" side of the fight, despite the misunderstandings. As a result, Woody understands that Americans, in general, are not "the enemy," and even those who misunderstand and mistakenly judge are still able to be shown how honorable people with Japanese heritage can be. If Woody's father had any doubts, they were smashed to bits when Woody decided to revisit Japan in order to connect to his roots. In this way, Woody's journey runs full-circle.