Is what the mother did, in taking out Zakat al-Fitr only for herself and her daughter, correct? And was it obligatory for the daughter to take out Zakat for herself, her mother, and her brother, or for the husband to take it out for his wife? And what is required if what happened was not correct?

You must pay Zakat (definition) al-Fitr for yourself. It is not obligatory upon the husband, brother, or parents, because you are responsible for your own sustenance. The husband is not obligated to provide for you until you have surrendered yourself to him and allowed him to enjoy conjugal relations; thus, Zakat is not obligatory upon him. Similarly, it is not obligatory upon the mother as long as she has money from which to spend. As for the brother, if he has money, then it is obligatory from his wealth; otherwise, it falls upon his mother if she is capable.