Tonight the heroes are tackling their first rumor. When they complete the rumor, do they just get the reward, or do they also get the 250 coins and 4 conquest from defeating the leader? I've found nothing saying either way.

It's still the boss of the third and final dungeon level, so I'd say so.

The one I'm iffy about is the Giant on the silver legendary dungeon, I think it mentions you get 4 conquest for killing him, although only the first time, but it says nothing of the 250 coins. We figured the heroes get the coins (only the first time of course), particularly since the previous lvl, while the third level, wasn't final, so the reward on there was only 100 coins and 2 conquest per boss (Gold beasts with the trial by fire dungeon, or whichever one it is with the master hound/dragon/demon. That dragon alone got the OL a TON of conquest). Rules don't make it especially clear however.

I played it that they got both rewards.

And it didn't matter, because they conceded the whole campaign. I obliterated them.