Is it permissible for a wife to go to the doctor or a resort without her husband's knowledge? And what is the ruling on a child refusing to live with the custodial mother?

It is not permissible for a woman to go out without her husband's knowledge, except out of necessity or need. This applies to visiting the doctor or going to a resort, provided there is no fear of temptation (fitna). The right of custody is not forfeited merely because the child refuses to live with the custodian. Rather, what is most beneficial for the child's welfare is prioritized, and the child's choice is not considered if they are of weak intellect.