Is a reunion considered legally valid after an initial divorce and receiving the divorce certificate, especially given that it occurred during the 'iddah period and without a new contract being written, and is the intercourse that took place afterwards considered a sin, and what is its expiation?

The waiting period (iddah) is not always determined by months. If a woman menstruates, her iddah is three menstrual cycles. If she does not menstruate, her iddah is three months. If she is pregnant, her iddah is until she gives birth. If the divorce was not the final and irrevocable third divorce, then the return (raj'ah) is valid if it occurs before the completion of the iddah, and there is no sin or expiation upon either of you. The return (raj'ah) is valid through a verbal declaration indicating it, such as the husband saying to his divorced wife during her iddah, "I have taken you back" (raj'atuki) or "I have returned to you" (irtaj'atuki). The right of return (raj'ah) before the completion of the iddah belongs to the husband and does not require the wife's acceptance, a new contract, or a new dowry.