Is it obligatory to fast for a week for someone who vowed to do so if a specific thing happened, and it did not happen? And is it permissible for a woman who has been irrevocably divorced for 16 years to return to her husband with a new contract and a new dowry?

Bain divorce is either a minor or a major bain. A minor bain occurs if a woman is divorced before consummation, or a revocable divorce whose waiting period has ended, or through khul' (divorce at the wife's instance), or by a judge's divorce for reasons other than an ila’ (vow of abstention) or inability to provide maintenance. In this case, the husband may remarry her with a new contract and mahr (dower). As for a major bain, it occurs after a triple divorce, and the woman is not lawful for him until she marries another husband and he consummates the marriage with her. If your sister's husband divorced her three times, she has become majorly bain from him, and she is not lawful for him until she marries someone else and he consummates the marriage with her, and then he divorces her. However, if he did not divorce her three times and described his divorce as bain, there is no harm for them to return to each other with a new contract, because his description has no effect.