In season 1 episode 10, why did the Slain Theocracy send the Black Scripture into E-Rental? I read that it had something to do with the World Class Item Downfall of Castle and Country but I don't know why they went to that location. Were they targeting Shalltear intentionally for the mind control or did they just happen across her?
No they where not targeting her, they had another target in mind and when they ran into her decided to target her instead because of the threat she posed to them. If I remember right it is explained in better detail in volume 3 of the light novel.
The Slane theocracy sent the black scripture on a recon mission to the country of E Rantel to both protect and use the world item Downfall of Castle and Country to prepare for the resurrection of the Catastrophe Dragon Lord. They happened to come across Shalltear after she killed the death bringing brigade and tried to use the item on her, however she was able to fight against the effects long enough to kill lady Kaide, the one who was using DOCAC. This resulted in Shalltear being mind controlled by a dead person, meaning that she was being controlled by no one. As such, she returned to her basic “Setting” if you can call it that I.e self defence.
From what I read it was a case of bad luck for both sides
The group was sent over behind the scenes as no one but them knew they were there. Similarly to how a country that sends an elite recon group to another country and then that country withdraws, leaving no trace. ainz knows they had to be someone powerful to have the world item, but he doesn’t know who. It could be E Rantel (unlikely cause ainz would know). Baharuth empire, (again unlikely cause ainz would know, so it is only a matter of time until Ainz finds out.