Is it permissible for a couple to reconcile after the husband divorced his wife twice while angry, and a third time at the officiant’s, knowing that one of the divorces occurred while the woman was menstruating?

If you were not aware of what you were saying at the time of divorce due to intense anger, then your divorce does not count. As for a divorce in the presence of an authorized official, it counts even if the wife is menstruating. According to this opinion, you can take her back before her waiting period (iddah) expires. However, if you were aware of what you were saying at the time of divorce for all three instances, then she is unlawful for you and will not become lawful for you until she marries another husband in a valid marriage and he consummates it, then divorces her.