What is the ruling regarding a father supplicating against his children without a cause, and his suspicion of them and his wife, and his demanding that she not pray or fast? Is it obligatory to obey him in that?

One must be kind to one's parents and associate with them in a good manner, overlooking their harm, even if they strive to make one associate partners with Allah. This is based on the Almighty's saying: (But if they endeavor to make you associate with Me that of which you have no knowledge, do not obey them, but accompany them in [this] world with appropriate kindness) [Luqman: 15]. Allah Almighty commanded righteousness towards parents and forbade disobedience to them, making disobedience one of the greatest major sins. One must be patient with the harm caused by parents, advise them, and supplicate for them, for Allah multiplies good deeds. Parents are also required to be just among their children.