Is it permissible to give Zakat al-Fitr from my own money on behalf of poor family members, such as my mother and sisters? And is it permissible to give it to my married brother who is experiencing difficult circumstances?

Zakat al-Fitr is obligatory for whoever is responsible for their maintenance. If you wish to give it on behalf of those whose maintenance is not obligatory for you, their permission is required, because the intention to give Zakat must come from the one on whom it is obligatory. As for giving Zakat al-Fitr to a brother, if his maintenance is obligatory for you, then it is not permissible. But if it is not obligatory, then it is permissible and even recommended, provided that he is among those who are eligible to receive Zakat.