What is the amount of zakat on gold that was sold after one year at a profit, and is zakat calculated on the purchase price or the selling price, and is it permissible to give it to relatives or an indigent brother?

Zakat is calculated at the price of the day it is disbursed, not the day of purchase. It is permissible to give zakat to poor relatives whose financial support is not religiously obligatory upon you. It is not permissible to give it to those whose financial support is obligatory upon you, such as ascendants and descendants (parents and children). The preponderant opinion is that the financial support of siblings is not obligatory, hence it is permissible to give them zakat.