It is the way of a man to search after a wife, and not the other way around - similar to a person who lost a possession. Who searches after whom? The one that lost his possession searches for it.
Why can betrothal be effected with money, and furthermore, why does the father of a bride who is a minor keep the money? From the law of a Jewish servant girl who goes free at twelve - “free, without money” - we know that there must be another case “with money” - that is, Kiddushin, which is in general effective with money.