Is it permissible for a girl to speak or smile at a young man she wishes to marry, even if the speech is merely a greeting and the smile is intended to show affection so that he feels her love and proposes to her?

Speech between unrelated young men and women without need is a gateway to fitna (temptation/strife). The Sharia came to block all means leading to haram (unlawful acts). Therefore, scholars prohibited a woman from greeting a man and a man from greeting a woman when there is fear of fitna. If a girl has an admiration for a young man, it is even more confirmed that she should be prevented from speaking with him without need, even if it is just to offer greetings, and more so to smile at him, in order to close the door to fitna and cut off the paths of Satan. However, there is no impediment for a girl to offer herself in marriage to a young man through a method that ensures the absence of fitna, such as the offer being made through some of her mahrams (unmarriageable kin).