Is it permissible to use a newborn's gift—that was given to him—to perform his aqiqah, and does this gift belong to the newborn or to the family?

The one obligated to perform the 'aqeeqah is the one responsible for the child's expenses, and he should pay for it from his own money, not from the child's money. If gifts presented on the occasion of the birth and 'aqeeqah are intended to assist the one responsible for the 'aqeeqah, then he may use them for the expenses of the 'aqeeqah. However, if they are intended for the newborn himself, or his mother, or others not obligated by the 'aqeeqah, then it is not permissible to use them for the 'aqeeqah except with the permission of the one for whom they were designated, and provided that his permission is valid. The recipient for whom the gift is designated is determined either by the donor's explicit statement or by common custom.