Is it permissible for a man to go with his family to perform Umrah this year, with an expiation for an oath, after he swore by divorce to his wife that he would not take them on any outings during this year due to his extreme anger, and would his oath be nullified by that?

There is no objection to performing 'Umrah (definition) with one's wife with the intention of worship, not recreation, because 'Umrah is an act of worship and not recreation. One must adhere to the oath not to take the wife for recreation for a period of one year, because the majority of scholars believe that an oath of divorce takes effect upon the occurrence of the condition it is contingent upon, in order to preserve the marital bond.