No. Such fanfiction is inappropriate as it goes against the general spirit and intended audience of the Percy Jackson series. The series is mainly aimed at young readers and fans, and this kind of content is not in line with the values and themes presented in the original books.
It could be a character who is jealous of the power that could come from combining the two worlds. Maybe a rogue demigod who wants to use Jessie's knowledge of her world's technology and the magic of the Percy Jackson world to rule over both. This character might start causing problems by stealing important artifacts from both worlds to try and build a device that would give them ultimate power.
Rather than seeking out such specific and potentially inappropriate fanfiction, I would recommend looking into fanfictions that expand on the world - building of Percy Jackson. There are some amazing ones that delve deeper into the mythology aspects, the backstories of minor characters, or create new and exciting quests for Percy and his friends.
The Percy Jackson series is mainly for young adult and teen audiences, and we should focus on positive and appropriate fan creations such as those about the characters' adventures, friendships, and growth, rather than inappropriate fanfic.
A really interesting plot would be if there was a prophecy in the Percy Jackson world that involved a 'stranger from a land of machines and modernity' which is Jessie. The two have to figure out how Jessie fits into this prophecy. Along the way, they face challenges like dealing with Jessie's lack of knowledge about Greek mythology at first and Percy trying to understand the strange technology Jessie brings with her.
Yes, there are many. For example, there are fanfics that focus on Percy's relationships with his friends like Grover and Annabeth in new adventures. Maybe they go on a quest to find a lost artifact in a new location that wasn't in the original books.
No. Fanfic should generally stay within the boundaries of good taste and respect. Extreme and inappropriate content like this goes against the spirit of the original work and can be offensive to many fans.
Yes, there are likely many out there. You can start by looking on popular fanfiction websites like Archive of Our Own. Just search for 'Jessie Luke and Jessie kiss' in the search bar, and you should be able to find relevant stories. However, the quality can vary greatly, so you might have to read through a few to find one that really suits your taste.
One possible plot could be that Jessie and Luke are friends who gradually develop deeper feelings for each other. They might go through some adventures together, face some challenges that bring them closer, and finally share a kiss in a moment of realization of their love. For example, they could be working on a project and start to notice little things about each other that they find endearing.