Which characters abide by romantic conventions for each gender and in what ways do they do this? Does abiding by gender conventions help or hinder romantic relationships? Why do the characters choose to abide by them?

Ivan Turgenev completed The Torrents of Spring, one of his greatest novels—and a work that's based, in part, on his own life story—in 1871. The story begins with the middle-aged main character, Dimitry Sanin, looking back at the past, so the bulk of the narrative occurs in the 1840s. A discussion of romantic conventions and gender roles must take this fact into account: social norms were different then.
In addition to the time frame, we should also note other details about the cultural setting as we think about conventions. The story takes place in Frankfurt, Germany, but the protagonist, Sanin, is Russian.
Now, back to the original question. The novel features a fairly wide cast of characters. We'll focus on a few of the key characters here, giving examples of how and why they conform to romantic conventions, and what effects those decisions have on their romantic relationships.
Let's start with Sanin. In an important moment that ends up defining his relationship with a young woman (Gemma) he's romantically interested in, Sanin behaves in a conventionally heroic way. This moment happens while a small group is having lunch at an inn, early in the narrative. A drunken officer stumbles up to the group's table, making a crude comment to Gemma about her beauty. The young woman is embarrassed and angry. The way that two of her male lunch companions react ends up being essential to the events that follow. Gemma's fiancé, Karl Klüber, tells the officer and his group to leave the restaurant. But Sanin goes much further in defending Gemma's honor: he berates the officer and gives him his own calling card, which would have been, in those days, practically an invitation to a duel.
In this case, it's Sanin that's behaving according to romantic conventions (he does, indeed, participate in a duel) and Klüber who isn't (well, at least not quite enough). This has an immediate impact on the course of the story. Gemma decides to end her engagement to Klüber, feeling that he didn't do enough to defend her honor. 
So what about Gemma, we might ask? How is she conforming to romantic conventions in this conflict? In some ways, she is behaving quite conventionally: she is angry at the inn but submissive, too, expecting her male companions to act on her behalf. Yet her choice to actually end her relationship with Klüber over this conflict isn't exactly conventional: she's being surprisingly proactive here. She's choosing her own happiness over what is expected of her, even knowing that she is causing a stressful situation for her mother, who worries that the engagement break will be viewed as a scandal.
Let's look at another example later in the novel, after the romantic relationship between Sanin and Gemma has been formalized. Sanin proposes marriage, Gemma accepts, and Sanin decides that he must sell his estate in Russia in order to properly provide for his future bride (all of these actions, we may note, are conventionally romantic on Sanin's part.) But during the business trip back to Russia, Sanin gets into trouble.
That trouble comes in the form of Maria Nikolaevna Polozov, a beautiful woman who's the wife of one of Sanin's old classmates. Despite his love for Gemma, Sanin is seduced by Maria Nikolaevna, and he ends up sleeping with her. Not exactly behavior that's becoming (or conventionally romantic) for a noble man who's worked so hard to achieve his dream of marrying the woman he truly loves. Matters are made worse when Sanin discovers that Maria Nikolaevna's desire for him wasn't even necessarily authentic: her seduction was part of a bet she wagered (and won) with her husband.
Maria Nikolaevna's behavior is the opposite of conventionally romantic: her aggressive and unsentimental actions and her open relationship with her husband would raise a few eyebrows even today. But she doesn't care. She goes back to her regular life, even as Sanin's life is ruined. He loses Gemma and lives on in deep regret: his failure to behave in a conventionally romantic way has been devastating to his personal life and to his overall happiness. 
We should note that many years later, at the end of the novel, when Sanin reaches out to Gemma, asking once again for her forgiveness, she behaves courteously. She's playing the conventional role of a wife to the man she eventually married, and she tells Sanin that she is happy with her life. In Gemma's case, following a conventional path has led to fulfillment and peace.