For the passage of time are both of these interchangeable?
What nuances does using either over the other have?
過ごす is clearly different from the other two. Its subject is usually a person (私は東京で1日を過ごした). It never takes a time length as a subject. (× 5か月が過ごす).
経つ and 過ぎる are very similar and usually interchangeable when the subject is a direct time length like 3日, 半年, 50年. In this case, 経つ is more objective, whereas 過ぎる may be slightly more emotional or nostalgic. When the subject is a noun that represents an event or a season, you can only use 過ぎる. For example you can say 夏休みが過ぎる／クリスマスが過ぎる but not 夏休みが経つ／クリスマスが経つ.