How does a woman who vowed to fast for four consecutive months calculate her fasting, given the interruption of the fast due to menstruation, and what is the ruling for one who unintentionally swallowed water during ablution: does she make up for that day or restart the months of fasting from the beginning?

The water that your sister swallowed unintentionally does not necessitate a make-up fast or a resumption of the fast, and her fast is valid.

As for the days of menstruation that interrupt her fasting, scholars hold different opinions on them: 1. She is obligated to make up the fast and offer expiation (kaffarah (definition)) (the relied-upon view among the Hanbalis). 2. She is obligated to make up the fast only, with no expiation due upon her (a view among the Hanbalis and Shafi'is). 3. Neither make-up nor expiation is due upon her (the more apparent view among the Shafi'is).

The closest opinion is that she is obligated to make up the fast without expiation, by analogy with Ramadan (definition), where making up the fast is obligatory due to an excuse, and expiation is not obligatory because she is excused.