Is a son obligated to obey his father who orders him to cut his hair, even though it is not very long or disheveled and there is nothing about it that necessitates cutting it?

You must obey your father in what he commands you, as long as it does not involve disobedience to Allah. Obedience to the father is obligatory if he has a valid reason and if there is no harm to you in it. So, if your father commands you to shave your hair because he sees in growing it an imitation of disbelievers or women, or because custom dictates that such an act is characteristic of ignoble people, then it is obligatory for you to obey him. However, if it is for no valid reason, then obedience is not obligatory.