Is it permissible to look into the eyes of a girl I love with the intention of marriage, without lust? And is it true that looking at what is not 'awrah in women is not problematic?

It is not permissible for a Muslim to intentionally direct his heart towards loving a particular girl. However, if this love occurs unintentionally and the soul abstains from what is forbidden, then there is no sin upon him. It is permissible for a Muslim to pray for the girl to be his wife, and if he is able to marry her, he should propose to her, because marriage is the best solution for those who love each other.

As for a man looking at a woman's eyes, it is not permissible, because Allah has commanded lowering the gaze. It is forbidden to look at a woman's face and hands with lust, by the consensus of jurists, and the preponderant opinion is that it is forbidden even without lust. If a Muslim is unable to marry the girl, he must divert his thoughts from her and focus on matters of his religion and worldly life.