Yes, there are likely many non - erotic fanfictions about them. A lot of fans focus on their friendship, adventures within the Blue Exorcist universe, and how they grow together as exorcists. For example, some might explore how they team up to face a particularly difficult demon, or how they support each other during training.
Their relationship within the context of the Blue Exorcist world is also a big draw. They are both exorcists in training, so there are opportunities for adventure and growth together. Fanfictions often explore how they support each other in battles against demons, and how their friendship might develop into something more. It's exciting to see how different writers imagine their bond evolving over time. Also, the fact that they are part of a larger group of characters in the series gives more scope for complex storylines in fanfictions.
You can start by looking on fanfiction platforms like Archive of Our Own (AO3). There are often a plethora of Blue Exorcist fanfictions, and you can specifically search for those featuring Rin and Shima. Another option is Fanfiction.net, which also has a large collection. Just use the search function and type in relevant keywords like 'Blue Exorcist Rin x Shima' to narrow down the results.
The theme of forgiveness could be explored. Maybe Shima makes a mistake that could put them in danger, and Rin has to decide whether to forgive him. Or vice versa. Additionally, their shared dreams of making the world a safer place from demons can be a central theme. They could work towards this goal in different ways and support each other along the journey.
Yes, the Blue Exorcist manga is over. It took readers on an exciting journey and concluded in a way that left a lasting impression. The final chapters brought closure to the characters and their adventures.