One thing I don't understand, why does the cardinal want shulo dead so badly?

I think she was just a chess piece to "spark" Something. Still don't know what he's trying to start tho....

The moment he saw shuli he felt the urge to own and to desire someone.
He thinks that desiring and wanting someone other than "god" was immoral and is a sinfull act. That's why he had always beat (you can see in a chapter where he was lashing himself) himself to atone for that sin.
He believes that Shuli is to blame for his desires for her.
Shuli is the one who "tempted him" thus she's evil and needs to be gotten rid off.

Fuck that's so messed up

Well not sure about that... He did want to spark something with her death, and that something is explained. He wanted to raise the power of the church, kill a noble and accuse some bandits so that the kingdom will have to turn to the church (whose leader belong to another country). Shuli was a perfect target because she's from a really old and powerful family and because of her reputation. The fact that he sees her as a temptress could have played too, after all he's really sexist. The cardinal is a really dark character.