Does a divorce issued by the husband while in a state of extreme anger, and by the wife while in her postpartum period (nifas), take effect? And is this divorce considered the third, after two previous divorces that occurred in a state of anger and during menstruation?

Anger alone does not prevent a divorce from occurring unless the person is unaware of what he is saying. The divorce of a woman in post-natal bleeding, like divorce during menstruation, is considered an innovated divorce (talaq (definition) bid‘ah), and it is valid according to the majority of jurists. Based on this, it is probable that your wife has become irrevocably divorced from you (baynunah kubra), and therefore, it is advisable to consult Islamic centers to clarify the appropriate fatwa (definition). One must be cautious of anger and strive to solve problems wisely, far from divorce.