What is my duty when my father asks me about my brother, knowing that my brother refuses that I speak about him? And should I be truthful with my father in my answer or lie to him? And is informing my father of my brother's true situation considered backbiting or فساد [corrupting] the relationship between them?

If your father asks you about your brother, tell him the truth and do not lie. This is because mentioning a person's faults is permissible if a benefit is realized, such as preventing a wrongdoing. Also, concealing the sins of a disobedient person is not always praiseworthy. So, if your brother will not be deterred except by informing your father, then inform him. Do not go along with him by emphasizing concealment, as that would encourage him to continue his actions. Rather, warn him that you will inform your father if he does not cease his wrongdoing.