Is it considered an act of disobedience (عقوق) to distance a mother who suffers from psychological problems and interferes in her children's lives, from living in the same house, while providing for all her needs? Is refusing to give money to a father who spends it on sins and boasts about it considered an act of disobedience (عقوق), despite his anger? And am I sinful towards my sister who cut ties with me because of a problem with her son, despite my repeated attempts to communicate and mend the relationship?

There is no blame upon you for your mother not living with you, as long as she is not harmed and fulfills her rights. This might even be better to avoid her irritability.

As for your father, you must honor and respect him. However, if you know that he will spend money on forbidden things, do not give it to him. Instead, substitute it with permissible gifts such as food and clothing. Apologize to him gently and kindly, and do not forget to advise him and pray for him.

Regarding your sister, you must maintain ties with her as much as you can. It is not permissible to sever ties with her even if she reciprocates with estrangement. The one who truly maintains ties is he who connects with his kin when they cut him off. Remember the noble Hadith: "If you are as you have said, then you are feeding them hot ashes, and you will continue to have support from Allah against them as long as you persist in that." And repel [evil] with that which is better.