Ppl don't usually use the word 닷새 in this sort of context, 5일 will be more adequate one. 닷새 is generally used in somewhat archaic conversations, and its use indicates the quantity of how many days have passed rather than the duration of time.
That is very true. People do not really use these days count like 닷새, 엿새, 이레, and so on.
What does the 번 mean here?
It means "times" as in "How many times"
It just means times. One time, two times, three times, etc.
Would using "동안" instead of "에" in this instance be incorrect?