Am I correct in supposing that Wave of Destruction can be used against spawn monsters? For example, it could be used to wipe out all three Riot monsters or all three Children of Abhoth? And furthermore, in the case of Riot monsters there would be no Sanity loss and no risk of the Terror level increasing?

I'm pretty sure they work with Spawn monsters as with any other monsters. But I'm curious to know about the other point. Wave of Destruction, Lightning Gun, and Flute of the Outer Gods, IIRC, all "defeat" monsters in the non-standard way.

Is "defeat" mechanically the same as "pass a combat check against"? Will you still get the piñata prize from the Warlock, Mi-Go, Spectral Hunter, Wizard Whateley, and Dunwich Horror? Will neutralizing a Cultist still add a doom token if you're against Yig?

I mean thematically, I don't see why the terror level wouldn't rise if you nuked a Riot with any of the above, so I presume that "defeat" does mean "pass a combat check against."

Here is something similar to your question, in regards to the Spawn Monster part of your question.

5. Do "return monsters to the cup" effects work on Spawn monster or no? Can the Feds Raiding Arkham actually get rid of the Dunwich Horror? (06/13/08)
No. Spawn monsters should never returned to the cup by card effects like "Feds Raid Arkham".

I believe that you still suffer the ill-effects of a Monster being removed from the board by other means then Combat in regards to an Ancient One/Herald's power, including Gate Closure.

I always interpreted "defeat" as treating the monster as if I had defeated it in combat; so I would get it as trophy, get the spoils from Mi-Go and like monsters, but also suffer adverse effects like those from Riots. Otherwise, it would probably say "return to the cup".

So "defeat"="pass a combat check against" in the same way that "delayed"="stay here next turn." Is that correct?

Well, we don't have an official ruling, but that will probably work for just about everything.