Well, in fanfiction, Klaus might kidnap Elena because he's lonely. He's been around for so long and has few people he can truly connect with. Elena represents something pure and good that he wants to possess. Also, he could be trying to use her as a pawn in his power games against other supernatural beings in Mystic Falls.
There are fanfictions where Elena and Klaus are thrown into an alternate universe. In this universe, they are the only two vampires left and have to figure out how to survive and repopulate the vampire race. This forces them to work together closely and, over time, they develop feelings for each other that are far from the hatred they initially had in the original Vampire Diaries storyline.
One idea could be that Klaus, after being away for a long time, returns to Mystic Falls and finds Elena in a difficult situation. He decides to help her out of a sense of an old, unspoken connection. As they spend time together, they start to understand each other better and a relationship begins to form.
One possible motive could be power. Damon might see Elena as a pawn in his game to gain more influence in the supernatural world. He could think that by kidnapping her, he can control those who care about her and make them do his bidding.
Sure. A lot of fans have written great fanfictions about Elena and Klaus' love. Some focus on the forbidden aspect of their relationship, like how Elena is supposed to be with the Salvatore brothers but has this unexpected connection with Klaus. It adds a new layer of drama and romance to the Vampire Diaries world.
Well, a prevalent theme is the idea of fate and how they are drawn to each other despite all the odds. Also, the theme of secrecy and the forbidden love aspect is quite common. Since Elena is part of the Mystic Falls gang and Klaus is an enemy in many ways, hiding their relationship from others is often explored in these fanfictions. And of course, there's the theme of sacrifice, where either Elena or Klaus has to sacrifice something for their love.
One possible motive could be his thirst for power. By kidnapping someone with a special ability or knowledge, like a powerful Force - user or a brilliant scientist, he could gain an edge in his quest to rule the galaxy. For example, if he kidnapped a Jedi who knew the location of ancient Sith artifacts, he could use those artifacts to increase his own strength.
One possible motive could be his fear of losing her. In the Star Wars universe, Anakin is deeply in love with Padme. If in fanfiction he feels that she is in danger or might be taken away from him by some external force or person, his desperation could lead him to kidnap her in an attempt to keep her safe. For example, if he suspects the Jedi Council of having some secret plan regarding Padme, his unstable state of mind might push him to this extreme action.
Well, Ursula could kidnap Ariel in fanfiction because she wants to take Ariel's place. Think about it. Ariel has a loving family, a great voice, and is adored by many. Ursula, on the other hand, is often shunned. So she might think that by kidnapping Ariel and getting rid of her, she can step into Ariel's life. Maybe she wants to be the one who is loved by Prince Eric. Or perhaps she wants to be the favorite daughter of King Triton. It's all about her twisted desires for a different kind of life.
It could be that Peter has some sort of knowledge or information that the kidnappers want. For example, he might have accidentally witnessed a crime and the criminals want to keep him silent.
She's likely to be defiant at first. Elena has a strong will, so she would probably resist Klaus's orders and try to escape.