What is the ruling on having many cats in the house, causing health, psychological, and physical harm to the household members and neighbors? Is the son who objects to their presence considered disobedient? What is the best course of action in this situation?

Your duty is to be dutiful to your parents and to please them with what is good, no matter how harsh their treatment, for Allah Almighty says: "And We have enjoined upon man [care] for his parents. His mother carried him with hardship upon hardship, and his weaning is in two years. Be grateful to Me and to your parents; to Me is the [final] destination. 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," and His saying: "And your Lord has decreed that you not worship except Him, and to parents, good treatment. Whether one or both of them reach old age [while] with you, say not to them [uff] or repel them but speak to them a noble word. And lower to them the wing of humility out of mercy and say, 'My Lord, have mercy upon them as they brought me up [when I was] small.'" If you do that, you will not be considered undutiful to them, and your reward will increase by enduring their harm. So be patient and seek reward from Allah, and change what you can without hurting their feelings, and entrust to Allah what you cannot.