Is it permissible for a woman to sell her zakat-eligible gold, most of which was bought by her father, and donate its price to her poor brother, and is she required to seek her husband's permission for that?

It is permissible for you to give charity from your money to your poor brother. By doing so, you will receive the reward for charity and for maintaining kinship ties. You are not required to seek your husband's permission if the money is less than one-third of your possessions. However, it is better to seek his permission to preserve affection.