In the 'Katherine kidnaps Elena fanfiction', it could be that Katherine has a grander scheme in mind. She kidnaps Elena because Elena is a doppelganger. Katherine might be working with some powerful dark forces who need Elena for a dark ritual. Elena would then have to fight not only to escape Katherine but also to stop this dark ritual from happening, with the help of her friends and allies who are desperately searching for her.
Perhaps Katherine kidnaps Elena out of jealousy. Elena has the love and attention of many people that Katherine once had or desired. So, Katherine might take Elena to a remote location, taunt her about all the things she doesn't have, and try to make her feel inferior. In this fanfiction, Elena could then try to outwit Katherine using her own wits and the help of her friends she manages to contact secretly.
The Salvatore brothers, Stefan and Damon, are the most likely candidates. They both have a strong connection to Elena and would stop at nothing to rescue her. Stefan's sense of chivalry and Damon's fierce protectiveness would drive them to track down Katherine and get Elena back.
One possible plot could be that Kol kidnaps Elena to use her as bait for his enemies. He might think that by having Elena in his custody, her friends, especially the Salvatores, would come to rescue her and he could then ambush them. Another idea is that Kol has some sort of vendetta against Elena because of her connection to the doppelganger bloodline. So he kidnaps her to extract revenge on the entire line.
One possibility is that Stefan is under some dark spell or influenced by an evil force. Maybe he kidnaps Elena to protect her from a greater danger that he believes only he can handle, but his actions are misinterpreted. For example, there could be a new vampire hunter in town who has a special weapon that can kill vampires instantly, and Stefan thinks the only way to keep Elena safe is to take her to a hidden location.
Jealousy could be one motive. Maybe Katherine is jealous of Elena's popularity and relationships, so she turns her into a vampire to make her life as difficult as hers. Another motive could be power. By turning Elena, Katherine might be trying to gain some sort of control over her or use her for her own plans.
It really depends on the perspective. Elena is often seen as the good girl, the one with a pure heart. She represents the moral compass in many fanfictions. But Katherine has this allure of mystery and a long history of survival. If you like characters with depth and a darker side, Katherine might be better in the fanfiction world. However, if you prefer the more traditional heroines, Elena would be the top choice.
Well, one possibility is that Elena's friends would immediately start a search party. Damon, for instance, would be furious and stop at nothing to find her. He might enlist the help of Stefan and Bonnie. They could use Bonnie's magic to try and track Klaus' whereabouts.
One possible motive could be Damon's possessiveness. In the 'The Vampire Diaries' universe, Damon has a complex relationship with Elena. He might kidnap her out of fear of losing her to someone else, like Stefan. Maybe he thinks that by taking her away, he can have her all to himself and make her see that he is the one she should be with.
One possible plot could be that Beetlejuice kidnaps Lydia to force her to help him gain more power in the afterlife. Maybe he thinks her unique connection to the living and the dead makes her a valuable asset.
One possible plot could be that Paul believes Bella has some special power or knowledge that he desperately needs. So, he kidnaps her in order to extract that information from her. However, as time passes, he starts to develop some unexpected feelings for her, which complicates the whole situation.
One possible plot could be that Harry is under the influence of a dark spell or some mind - controlling magic that makes him kidnap Ginny. Maybe he's been hexed by a powerful dark wizard who wants to use Ginny as a bargaining chip against the Order of Phoenix. So, Harry, without his true self in control, takes Ginny to a hidden location.