From Rulebook page 11 wrote:
When a monster is defeated by a hero, it is killed and Yes, when Dorgan defeats a Roaming Monster due to one of his traps then that Encounter goes into his loot pile.
that hero places the monster in his loot pile. When a
monster dies outside of battle, it is placed in the loot
pile of the player who caused it to die.
Now, the trap states it only does damage to those monsters who are roaming to it, so if someone puts the trap on, let's say The Den tile, and a monster is generated there my guess is that the new spawn is not affected by this trap. But this is only an assumption. Good Luck.