They could develop a shared hobby. For instance, they both get into painting and use it as a way to express their emotions and connect on a deeper level. It would be unique because it shows a side of them that's not related to their supernatural natures.
I'm not sure specifically as it depends on the particular fanfiction. It could be a story where a character named Kol kidnaps Bonnie within the universe of the show or book that the fanfiction is based on. Maybe it's part of a complex plot involving magic, power struggles, or personal vendettas.
Elena would likely be angry. She has her own agency and doesn't like being controlled. If Stefan kidnapped her, she would fight back. She might use her knowledge of the supernatural world and her relationships with other characters to try to escape. For example, she could try to call out to Damon telepathically if they have some sort of connection in the story.
One popular 'kol and elena fanfiction' might be those that explore their relationship in an alternate universe where they are on a journey together to discover a hidden magic. In these stories, Kol could be protecting Elena from various threats, adding an element of adventure.
The relationship would shock Damon and Stefan. They have always been protective of Elena, and Kol has a reputation as a dangerous vampire. Damon might be especially angry and try to drive Kol away from Elena. Bonnie would also be concerned as she has seen Kol's past actions. However, Caroline might be more open - minded and could potentially try to help Elena understand her feelings for Kol. Klaus would be an interesting factor too. He might see Kol's relationship with Elena as a way to gain an advantage in his power games or he could be protective of his brother's new - found love.
You can try fanfiction websites like Archive of Our Own (AO3). It has a large collection of fanfiction from various fandoms, and you're likely to find some great 'kol and elena fanfiction' there.
One possible storyline could be that Kol and Elena's relationship starts in a rather unexpected way. Maybe they are forced to work together to defeat a common enemy in the supernatural world. During this time, they gradually develop feelings for each other. And when they find out Elena is pregnant, they face a lot of challenges. Some vampires might oppose their union because of Kol's past actions. But they decide to fight for their family and the future of their baby.
One possible plot could be that Kol and Elena meet by chance in a place outside of Mystic Falls. They initially clash due to their different personalities and pasts. Kol is intrigued by Elena's tenacity and her kind heart despite all she has been through. As they are forced to spend more time together, perhaps because of a common threat, they start to understand each other better. Elena begins to see Kol not just as one of the bad guys but as someone with his own complex story. Their love could develop gradually as they share their secrets and fears, and finally, they realize they are in love.