The plot could involve the Phantom seeing Christine as his only salvation from his lonely and tortured existence. So, he kidnaps her with the intention of making her understand his world. He may show her his musical creations, hoping she'll come to love him in return. However, Christine might be initially terrified and try to escape.
Another possible antagonist could be the new opera managers. They could be trying to use Christine's disappearance to their advantage, perhaps by spreading rumors that she has run away with a lover to create a scandal and boost ticket sales. They might also try to stop Raoul from searching for her, seeing it as bad for business.
Fanfiction about them often delves into the unspoken emotions. It can show the Phantom's vulnerability more clearly. Maybe in a fanfic, he reveals his past traumas to Christine, which deepens their bond.
The relationship can start with mystery. For example, Christine could hear the Phantom's voice in the opera house but not see him at first. Then, as she gets closer to uncovering his identity, there's a sense of both fear and attraction. This could lead to a series of meetings where they talk about music, and Christine slowly begins to see the man behind the mask, both literally and metaphorically.
One possible plot could be that Erik, being the complex and often misunderstood character, kidnaps Christine out of his obsessive love for her. He might take her to his lair, thinking he can protect her from the outside world which he believes is full of people who don't truly appreciate her talent. Then Christine could struggle between her fear of Erik and her understanding of his tortured soul.
Yes, there are many. Some fanfictions focus on their emotional connection, like how they first met and the gradual development of their understanding and respect for each other. For example, there are stories where Christine discovers the Phantom's tragic past and tries to heal his inner wounds through friendship.
There are many positive alternatives. For example, you can explore fanfiction that focuses on the deep emotional bond between Christine and the Phantom in a more respectful way. Some fanfictions might explore their relationship in a platonic or romantic but chaste sense, delving into their shared love for music and the opera house.
I'm not aware of any such fanfiction. Fanfiction should ideally be in line with ethical and moral standards, and creating 'naughty' or inappropriate fanfiction is not acceptable behavior in the fan community.
One common theme is redemption. The Phantom, often seen as a tragic figure, may seek redemption through his relationship with Christine. He might change his ways and learn to love in a more positive and selfless manner.