What is the ruling of Islamic law regarding the permissibility of a wife returning after being divorced three times, and can a judge reinstate her based on the laws, especially with the claim that the second divorce occurred while she was pregnant, or that her name was not explicitly mentioned in the previous divorces?

Divorcing a pregnant woman is valid. Your brother's triple divorce of his wife means he cannot take her back unless she marries another man, consummates the marriage with him, and then he divorces her or dies, and her waiting period expires. The custody of the children belongs to the mother if she is trustworthy with them. It is not permissible for the custodian to prevent the other party from seeing the children. In case of dispute, the matter should be referred to the Sharia (definition) court.