All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
If the mentioned non-Muslim girl belongs to the People of the Book (Jew or Christian), then it is permissible for you to marry her provided she is chaste. Allah Says (what means): {(Lawful to you in marriage) are chaste women from the believers and chaste women from those who were given the Scripture (Jews and Christians) before your time, when you have given their due Mahr (bridal money given by the husband to his wife at the time of marriage).} [Quran 5:5]
Marrying her in the mentioned circumstances would be a reason to encourage her to embrace Islam. In spite of her desire to embrace Islam it is not compulsory for this brother to marry her if he is not interested in her. If he does not like her and he marries her only to reconcile her to adhere to Islam, it might end in divorce, and subsequently, it may cause hindrance for her from adhering to Islam.
We would like to inform you that the most important condition to marry women from the People of the Book is chastity which is rarely found among them nowadays. Moreover, a lot of them are atheist and do not adhere to any religion though the community in general is considered as Christian or Jewish.
Even if she is a practicing and chaste Christian or Jew, consider that there are many spinsters in the Muslim community who are more desirable to marry than persons of other religions.
For your second question, please refer to Fatwa 82189.
Allah knows best.