What would happen if Lisbeth Salander and Sherlock teamed up to solve mysteries?
At first, they'd hate each other, and hate working in a team. They'd have different approaches to solving crime.
Lisbeth would take tech-oriented/cyber crimes cases with a specific moral line, especially those to do with the abuse of women and girls. She wouldn't suffer Sherlock's ego, and would ghost on cases that didn't interest her. Although she's highly intelligent, she doesn’t do Sherlock's rapid observation deduction thing (which is more performative anyway). But she is meticulous.
Sherlock would throw his weight around trying to prove that he's the smartest person in the room. He'd inadvertently insult her (or sometimes deliberately, to get a reaction from her). There'd be friction at first, but they would challenge, and ultimately respect each other (and bond over a mean game of long-distance chess).
Because they're both kind of "larger than life" figures, I don't see them sharing the same story. There's not enough oxygen in the room, as it were. Or if they did, they’d have to have an impartial buffer, like Watson (but without the bias), who could interpret and humanize each to the other, and can act as a go-between when they’re being stubborn. Otherwise, I think the two could be those interesting consultants that people like them always have in their orbits, to pull out at the 11th hour, for those wonderful hail Mary passes. How often have you seen them say something like "I have a friend in [London] who might be able to help." They could each be that friend!
Also, it would be interesting to see the nuances in their different social disorders, and how the two would relate to one another.