In the BSD manga, unfortunately, Dazai did die. His death was a crucial moment that influenced the direction of the story and the relationships among other characters.
One common theme is identity exploration. Since Dazai is a complex character, when he crosses over into different universes, he often has to re - evaluate who he is. Another theme is the clash of different worlds' values. For example, if he crosses into a more morally strict world, his rather ambiguous moral compass creates interesting conflicts.
One great 'bsd dazai crossover fanfiction' is 'Dazai in Another World'. It takes Dazai from BSD and places him in a fantasy setting. The story plays with his character traits in a new and interesting way, exploring how he'd interact with the new world's rules and characters.
Well, since it's related to Bungo Stray Dogs, perhaps it's about a special entrance exam within the agency or some other organization in the story. Dazai is a complex character, so it could involve his schemes or his interactions with other characters during this exam. It might also explore the rules and challenges of this exam, which could be unique to the world of BSD.
One common theme is power exploration. Since both BNHA and BSD have characters with unique abilities, fanfics often explore how these powers interact and combine. For example, a character's quirk from BNHA might be used in an unexpected way when paired with a power from BSD.
One characteristic is the exploration of their complex relationship. They are often portrayed as having a love - hate relationship. So in many fanfics, there will be a lot of tension between them.
There's also 'Jujutsu and Bungo in an Alternate Realm'. This fanfic delves deep into how the powers from both universes would interact. It shows the characters trying to understand each other's abilities and form alliances. It's a long read but definitely worth it for fans of both series.