There might be a fanfic that focuses on their early days at the firehouse. It could explore how they first met, the initial impressions they had of each other, and how they gradually became the close friends and reliable colleagues that we see in the show. Maybe there were some misunderstandings at first that added drama to their relationship development.
One of the common themes is definitely the exploration of their individual characters. Fanfics might dig deeper into Stella's past experiences and how they shape her actions towards Severide. Similarly, Severide's character traits are often examined more closely. Also, the theme of loyalty is quite prevalent. In the context of the firehouse and their relationship, loyalty to each other and to their colleagues is a big part of many fanfics. For example, there could be a situation where one has to choose between loyalty to the other and loyalty to the firehouse rules, and how they handle that makes for an interesting plot. And of course, the theme of family is there too. The firehouse is like a family, and Stella and Severide's relationship is also part of that family dynamic, which is often explored in fanfics.
You can try fanfiction.net. It has a large collection of fan - created stories, including many for Chicago Fire. Just use the search function and type in 'Chicago Fire Casey and Severide' and you'll likely find a good number of relevant fanfics.
Well, I'm not entirely sure about the most up - to - date ones, but some classic 'Chicago Fire Stella and Severide' Tumblr fanfics are really great. For example, there are those that explore their backstories and how they came to be the people they are in the show. Fanfics often add extra layers to their relationship, like the one where they have to deal with an external threat together and how that strengthens their connection even more. Some fanfics also play with different 'what - if' scenarios, like if one of them got injured in a different way than in the show and how the other would react.
No, it is not appropriate. Fanfiction should generally adhere to ethical and moral standards, and writing about sexual content in relation to a show in this way is not acceptable. It can also be considered a form of inappropriate and disrespectful use of the original work.