Is a son's charging of his father's phone—which the father uses for unlawful matters, such as viewing the doubts of the Khawarij and Ash'aris, or pictures of women—considered aiding in sin?

You must obey your father in what he commands you, as long as it does not involve disobedience to Allah Almighty or harm to you, even if he sometimes uses the phone in ways that are not Islamically permissible. Your action in this regard is not considered assisting him in wrongdoing. If you see him using the phone for objectionable matters, supplicate for his guidance and advise him gently and kindly.