Does a divorce take place if the wife returns home after her divorce was made contingent upon her leaving, given the husband's intention not to effect a divorce? And is it obligatory to renew the marriage contract with a dowry and a guardian after the husband took her back and paid an expiation for an oath?

Marital problems are resolved through dialogue, understanding, and forgiveness, not through divorce. Regarding the suspension of divorce: if the wording of your oath entails the occurrence of divorce merely upon her leaving without you, then the oath has been violated. According to the majority of scholars, if the oath is violated, and you intended divorce, or to prevent her, or you did not intend anything, and the wording of the oath was explicit regarding divorce, then the divorce occurs. Your taking her back before the completion of the waiting period is valid if this divorce does not complete the three divorces. Sheikh Al-Islam Ibn Taymiyyah holds that whoever intends by an oath to urge or to prevent, then upon violating the oath, he is obligated to pay an expiation for an oath. In this case, if you doubt your intention to effect the divorce, then the original state is the continuation of the marriage. You are permitted to follow the opinion of Ibn Taymiyyah, unless it is for the purpose of seeking concessions.