Yes, there are quite a few. You can start by looking on popular fanfiction websites. Some might focus on their relationship dynamics in the context of the original story's world, exploring how their different personalities interact.
Another option is Tumblr. There are often fanfic writers who post their works there, and you might be able to find some Himeno x Denji fanfiction by searching relevant tags or following the right blogs.
No. Such stories are not acceptable in literature as they are vulgar and violate the norms of good taste and decency. Literature should aim to promote positive values, emotions and ideas.
Since 'amy fleming pukes fanfic' is not a widely known mainstream work, it's difficult to give a detailed account. It could be that the author of this fanfic wanted to explore a more gritty or real - life aspect of the character Amy Fleming. Puking could be used as a device to show her vulnerability, perhaps she's been poisoned or is extremely ill due to some event in the story's universe. It could also be part of a comical or dark - humored take on the character, where puking is just one of those unexpected and somewhat gross elements added for shock value or to make the story stand out in the realm of fan - created content.
Well, it could be a number of things. Maybe she faces some challenges, has a major life change, or discovers a hidden truth. But without reading the specific novel, it's hard to say for sure.