Must a vow of four rak'ahs daily be fulfilled, which was vowed to escape an imagined calamity that later proved to be non-existent, and can three days of fasting be observed as expiation for it instead of fulfilling it, knowing that the one who made the vow was a child?

So long as the situation is as you described—that you were young at that time—your vow is not valid and nothing is binding upon you. This is because a vow is only valid when made by an adult. Thus, the vow of a child who has not reached puberty is not valid. The one making the vow (the nadir) must be an adult, Muslim, acting by choice, and capable of disposing of what he vows. Therefore, a vow made by someone who is not an adult, such as a child or an insane person, is not valid.