Is what is happening to my friend—that she sees the specter of a specific person and talks to him while awake, and he urges her to do good deeds, and makes her affected by what befalls him—something normal, and is thinking about it considered forbidden?

A man's heartfelt inclination is not problematic, but one must be careful not to excessively indulge in it lest it lead to infatuation. There is no harm for a woman to offer herself in marriage to a righteous man, within certain guidelines. As for the symptoms experienced when imagining a person, one should seek refuge in Allah from Satan and maintain regular remembrance of Allah (dhikr).