When a hero defeats a monster and it becomes the overlord's turn, if the overlord places that monster back on the board where it was defeated, is the monster activated and can the overlord use it during that round that it was placed?
I would vote for: Yes!
I think the real question is, can the Overloard place it directly on his turn, or does he have to wait until the next round? I think there is nothing in the rules that presses him to wait. I think the rules for reinforements are clear about when they occur, they are usually given in the scenario description.
If there is a monster on the board, the Overlord can activate it. There is no summoning sickness.
If the Monster is killed in current Hero turn, the Overlord can spawn it again in his turn. He just need to watch the groupsize and the numbers of Monsters he's allowed to spawn and the Moment.
(And yes it can be an Advantage for the Overlord, if you kill an single big Monster and he's allowed to spawn it in the beginnig next to you. Its like an free heal. But thats descent and an tactical component. (Our Overlord Killed his Monster with a trap, to get a new on with full hp.))