Is a wife considered to have returned to her husband after her verbal and written reinstatement, given that a document proving the رجعة (return to marital status) was not obtained from the court, especially since she believes she is still divorced due to the absence of an official document to that effect?

As soon as the wife is reconciled with during her waiting period (iddah) following a revocable divorce, she returns to the marital bond of her husband, and she is not permitted to refuse. The validity of the reconciliation does not depend on a document, but it should be obtained to preserve rights and prevent corruption.