Is the aforementioned "Rajah" (return to marital union after a revocable divorce) considered valid, and is there such a thing as conditional divorce, especially after the wife waives her rights in the divorce paper?

A husband's marriage to a second wife does not permit his first wife to demand a divorce. If the divorce occurred by mutual release (ibra’), meaning the wife relinquished her rights in exchange for the divorce, then this is an irrevocable divorce (talaq ba'in) where the husband does not have the right to take his wife back during her waiting period ('iddah). This is a type of khul'. However, if the divorce occurred without compensation, and then the wife relinquished her rights after the divorce took place, then this is a revocable divorce (talaq raj'i) where the husband has the right to take his wife back as long as she is within her waiting period, and her consent is not required, based on the Almighty's saying: "And their husbands have more right to take them back in that [period] if they want reconciliation." In case of dispute, it is advisable to seek arbitration from people of knowledge and religion.