Is repeating supplications or performing acts of obedience due to obsessive-compulsive disorder (OCD)—especially when fearing for family and redoing ablution—considered a religiously permissible solution, and is there a specific time for repeating supplications, or is there a fundamental solution for this OCD?

There is no specific number for supplicating to protect one's family and entrust them to Allah. However, the Prophet, peace and blessings be upon him, when he supplicated, he would do so three times, and when he asked, he would ask three times. It is recommended to repeat supplications three times. There are also specific supplications reported from him, peace and blessings be upon him, for protection and well-being, to be recited morning and evening, some of which are said once, and some are repeated three times. As for ablution (wudu'), the obligatory act is a single wash, and one should not pay attention to whispers (waswasah) after being certain.