Does the behavior of things-that-behave-quantumly typically affect macro-level events...?

Yes. They only appear weird if you look at small enough scales, but classical electrons would not have stable orbits, so without quantum effects there'd be no stable atoms.

Is there some way to prove that quantum events are random, as opposed to caused deterministically by something we just haven't found?

No, but there is evidence. There is a proof that if they were caused by something unknown but deterministic (or if there even was a classical probability function for certain events) then they would follow Bell's inequalities. But that appears not to be the case.