Is expiation (kaffarah) obligatory for the oath, "By God, I will never bring her into the house again," after the friend arrived, knowing that the oath was uttered in a state of extreme anger?

Quarreling with one's mother and raising one's voice at her is forbidden and considered an act of disobedience (ʿuqūq). Your statement, "By the life of Allah, I will never bring her to the house again," is considered a binding oath. If your friend comes [to your house], you are required to offer expiation for breaking an oath.