Jealousy is a big one. Karin might be jealous of Sakura's relationship with Sasuke. So, she frames Sakura to break them up or make Sasuke see Sakura in a bad light.
Karin may also frame Sakura out of a sense of self - preservation. If Sakura discovers something Karin has done wrong or is a threat to Karin's plans, Karin could frame Sakura to divert attention from herself and make Sakura the scapegoat instead.
Another motive could be competition. If there's a competition for a powerful position or a rare resource in the fanfiction world, Karin may frame Sakura to eliminate her as a competitor. For example, if there's a chance to study under a great master, Karin might frame Sakura to get that opportunity herself.
Often, Sakura is depicted as kind - hearted. She is the one who is more likely to see the good in people. Karin, on the other hand, might be more headstrong. For example, when they face a difficult situation, Sakura will try to find a peaceful solution first, while Karin is ready to jump into action right away.
The theme of cooperation also pops up a lot. Since they have different abilities, fanfictions often explore how they can work together. For instance, Sakura's knowledge of herbs and healing combined with Karin's knowledge of the chakra flow can lead to some interesting storylines where they solve problems or face enemies as a team.
Power struggles are also quite common. If they have different abilities or powers in the source material, the fanfiction might explore who is stronger, how they use their powers against each other, and what they can learn from their battles. It could also be about proving one's worth, especially in a world where power and status matter a great deal.
A common theme is competition. Since they are both characters with their own strengths, fanfictions often center around them competing for things like recognition, a certain goal, or the attention of other characters.
Love triangles are a very common theme. Sasuke is often at the center of the attention of both Sakura and Karin, which creates a lot of tension and drama.
In the realm of fanfiction, almost any scenario is possible. Regarding Sasuke divorcing Sakura for Karin, it may exist. Fanfiction authors draw inspiration from the source material and then take it in new directions. The idea might stem from Karin's past feelings for Sasuke and some fans' desire to see how such a relationship shift would impact the characters. It could also be a way to explore the characters' personalities more deeply, for example, how Sakura would cope with such a betrayal and how Karin would handle being with Sasuke in that context.
There probably is. Fanfiction writers like to take existing characters and create different storylines. But it's important to note that this isn't part of the official Naruto story. In the original, Sasuke and Sakura are a couple, and Karin has her own relationship with Sasuke which is more of comradeship in the official plot. Fanfiction just gives people the freedom to deviate from the original.
Karin could be heartbroken. She might initially be angry and feel betrayed, lashing out at Sasuke for choosing Sakura over her.
One common theme is the exploration of unrequited love. Sakura's love for Sasuke is well - known, and when he leaves her for Karin, it delves into the pain of unrequited love from Sakura's perspective. Another theme is the idea of choosing based on power and usefulness. As Karin has certain powers that can be of help to Sasuke, it shows a theme of practicality in relationships.
I don't have specific knowledge about the common themes in 'karin x sakura lemon fanfic' as it depends on the creativity of the writers. But friendship or adventure could be possible themes.