What is the ruling on a brother cohabiting with a Christian woman and having a child with her without marriage, under the pretext of financial inability, knowing that he is unemployed and his mother gives him money?

The brother's cohabitation with a woman without a legitimate marriage contract is considered zina (fornication/adultery), and it is one of the gravest major sins. It is not permissible for your mother or anyone else to assist him financially as long as he insists on committing this forbidden act. Rather, it is obligatory to forbid him from this evil and to help him enter into a legitimate marriage. The child resulting from this forbidden relationship is a child of zina and is not attributed to his father, nor do the legal rulings such as mahramiyyah (prohibition of marriage due to kinship) and inheritance apply to him.