Is the wife's divorce valid, and does the oath of divorce take effect if the husband utters it while knowing that she is menstruating?

The phrase "oath of divorce" is general. If the word you uttered was an explicit, unconditional divorce, such as "You are divorced," then divorce occurs, even if you intended reconciliation. If it was a euphemism and you did not intend divorce, then it does not occur. If it was an oath by divorce, then its ruling is the same as that of a conditional divorce, and the preponderant opinion is that it occurs upon the fulfillment of the condition.

The preponderant opinion with us is that divorce during menstruation occurs. It is most advisable to consult trustworthy scholars to ascertain the true nature of the word you uttered.

If divorce occurred and it was not the third divorce, then the reconciliation (rujūʿ) is valid. The wife's consent is not a condition for the validity of reconciliation; rather, it is required that the return occurs before the end of the waiting period (iddah).

If the wife leaves the marital home before completing her waiting period from a revocable divorce, she has erred and sinned. Similarly, if she leaves during the waiting period without necessity. She is disobedient (nashiz) by leaving the house without permission, and you have the right to prevent her until she ransoms herself, unless she has a legitimate Shari justification, in which case it is not permissible to harm her to make her ransom herself.