In this case the church do a full on investigation just because of a rumor, and it looks like everyone knows the power struggle between the church and the empire. The emperor and the dukedom can use it to keep the church in check or spread rumors about the church using excessive force for nothing
Nah, if police does a raid based on loosey goosey rumors they would be mocked and probably lose their jobs, at least in most modern western countries. Police cannot invade private property without prior approval, and that is usually only given with concrete proof, which means that a simple rumor wouldn't be enough. Even if the rumor turns out to be true, the police's neglegence to follow protocol could ruin the whole case and allow the criminals to get away scotfree.
I dont see how the church would become mocked after the investigation turned up nothing, they were just following up on rumours, just like how police follow intel. They aren't mocked if the intel turns out to be false