Is it permissible for me to leave my family's home after sacrificing my money to avoid family problems?

If the only reason for leaving your parents' home is what you mentioned, we do not advise you to leave due to the potential for estrangement, ill-will, and increased problems that may result. You have done well by sacrificing to preserve your father's affection and dutifulness, and your brother's connection and love. We ask Allah to grant you abundant reward and compensate you with something better, for "You will not abandon anything for the sake of Allah, the Mighty and Majestic, except that Allah will replace it with something better for you."