Is what transpired—a marriage proposal in the presence of family and witnesses, and acceptance from both parties, without mention of the dowry—considered a legally valid marriage?

The mention of the dowry is not a condition for the validity of the marriage contract. However, the formula mentioned does not constitute a valid marriage contract, because the statement: "We have come to seek your daughter's hand in marriage" is merely a proposal. As for the marriage contract itself, it requires a form of offer and acceptance, such as the guardian saying: "I marry you to so-and-so," and the husband saying: "I accept her marriage." Therefore, what transpired between your brother and the girl's father does not constitute a valid marriage.