What is the ruling on the paper that was written between my brother and a girl, believing it to be a marriage, and what is the ruling on the subsequent legitimate marriage that took place between them?

What happened with your brother and that girl, the writing of a paper between them—like a marriage contract—before the lawful (shar'i) contract, does not make the forbidden permissible, nor is it a lawful marriage. Rather, it is deception, fraud, and an attempt to circumvent God's prohibitions. It lacks the conditions of a valid marriage, namely the guardian, the two witnesses, and the dowry. The contract that entails lawful rights is that which occurred thereafter.