Is a marriage that took place without the guardian's approval—due to his refusal of the dowry—whereupon the girl proceeded to marry the young man in the presence of his sister, the religious officiant, and witnesses, and then the girl's family found out about it and later approved after they had children, considered a legitimate marriage or not, and what is the ruling on this marriage?

This marriage is not Islamically valid due to the absence of the guardian's (wali) consent. The marriage contract (nikah (definition)) is void without it and must be annulled, even if the couple has had children or if the guardian later gives his approval. However, if a judge from a school of thought that does not stipulate the guardian's consent rules it valid, then it is considered valid. To remarry, the guardian's consent is required, along with the fulfillment of all conditions for a valid marriage contract.