What is the ruling on becoming involved with a non-Muslim girl, and is it permissible to supplicate for her guidance or invite her to Islam with the intention of marriage?

Acquaintance between foreign men and women is a gateway to temptation. If the girl is not from the People of the Book, it is not permissible to marry her, based on the verse: "And do not marry polytheistic women until they believe." If she is from the People of the Book (Jewish or Christian) and chaste, it is permissible to marry her through her guardian, but it is preferable to marry a Muslim woman. As for inviting her to Islam, that is a good deed, and it should be done through righteous Muslim women to avoid temptation. Loving a disbeliever or a sinner with a natural, inherent love does not diminish faith, as long as one dislikes the sin or disbelief that the person holds.