Is a wife's accusation of adultery against her husband in front of his family considered an act of slander, and is she entitled to seek divorce based on her claim of his betrayal, and what is the fate of the children, their maintenance, and their housing in the event the husband refuses to divorce and threatens to file a slander lawsuit?

It is forbidden for a Muslim to accuse another without clear proof. Conjectured circumstantial evidence is not sufficient. Proof is established by confession or the testimony of four men. If the accuser does not provide proof, the accused has the right to demand the punishment for false accusation (hadd al-qadhf) by proving the accusation with two witnesses. The mother has a primary right to the custody of the children after divorce, as long as she does not remarry and meets the stipulated conditions. The children's maintenance is obligatory upon the father.