What is the ruling on divorce without intention, and does it take effect if the husband utters it in a state of extreme anger, given a prior intention to divorce on a previous occasion, and his desire to reconcile with his wife to protect the family?

An explicit divorce takes effect the moment it is uttered and does not require an intention. Severe anger does not prevent its occurrence as long as the speaker is aware of what he is saying.

If you utter the divorce statement three times, attributing it to your wife, then three divorces are binding, even if one of them was without intention, provided that the divorces occur while the wife is still under your marital bond.

However, if the phrasing is ambiguous, then divorce is binding only with intention.

You can reconcile with your wife before her 'iddah (waiting period) is complete, provided that the divorce has not been completed three times. But if it has been three times, then she is irrevocably divorced from you (baynunah kubra), and she is not lawful for you until she marries another man and he consummates the marriage with her, and then he divorces her. It is advisable to consult trustworthy scholars of knowledge to clarify the details.