Rule 3 mentions that you must send at least one error or crash event through the Backtrace API in order to be elegible, which makes sense, but it's not clear whether that can be a "Hello World" sort of event at the beginning of the game or if it has to be an actual error. What if the game doesn't error at all?

It can be anything, even a test "Hello World" Debug.LogError or something.  The goal is to just ensure that you have actually integrated with Backtrace.  Once one error / anything is sent by your game and received by Backtrace, that means it is all set up and the integration is working.  From there, you dont have to touch the Backtrace SDK through the rest of the jam, it should just automatically send error reports without you needing to do anything.