Does the divorce upon which I swore to my wife while she was menstruating, and which I did not carry out after she became pure, despite my firm intention, take effect? What is the nature of رجعة (reconciliation/taking back) in the Sunnah? Do my family have rights over my wife, and do I have rights over her family? What are these rights?

Divorce does not occur merely by the husband swearing to his wife that she will not remain married to him, because this is a promise of divorce, and a promise of divorce is neither binding nor recommended to be fulfilled. Breaking this oath only occurs when there is despair of it happening, which would be shortly before death. The family does not have a special right over the wife or vice versa, but maintaining kinship ties is obligatory. Kindness and appreciation should be shown between in-laws, following the example of the Prophet, peace and blessings be upon him.