Is the aforementioned 'Urfi marriage Islamically valid, and what is the status of the child born from it?

Marriage without a guardian (wali) is invalid, and the spouses must separate. The woman must observe the 'iddah (waiting period) due to cohabitation within that marriage. If the husband wishes to marry her again, it must be after her 'iddah has concluded, and her guardian should then عقد (contract) the marriage for her, fulfilling all the conditions and pillars [of a valid marriage]. As for the child, she is attributed to the husband and is not considered a child of zina (fornication), as long as the spouses believed their marriage was permissible. Any marriage in which the husband believed it was permissible and engaged in cohabitation therein, his child born from it is attributed to him, and they inherit from each other by the unanimous agreement of Muslims.