Is it valid for a husband to take back his wife if he divorced her more than three times in a state of extreme, unconscious anger, without intending divorce, especially given that the wife confirmed this and did not leave the marital home until after the final divorce?

If one's anger reaches a level where he no longer comprehends what he is saying, then his divorce is not valid. Therefore, the questioner's divorce is not valid, and he can continue to live with his wife. Divorce uttered with explicit wording does not require intention. It is advisable to consult a Sharia court.