Hero is the guy who has Aura 4 on the overlord's turn, and he has the skill that gives him necromancy. If a monster is killed by his aura, can he use necromancy on the killed monster?

I would say yes, because all necromancy says is "killed by (the hero)", not "killed as a result of an attack by (the hero)" Really, aura is still the hero doing the damage, though not via an attack.

I'd like to hear opinions on this though, since I'm far from certain.

I'm just not sure on this one. Using the RtL rules for both it gets a little fuzzy.

Necromancy
When a hero with the Necromancy ability deals the killing blow
to a small (no bigger than one space), normal, unnamed monster
(and it stays dead, in the event of Undying or other such effects)
he may choose to animate the monster. The monster stays on the
board and is returned to full health, but falls under the control of
the hero. A hero cannot control more than one monster at a time,
but may choose to let a monster under his control die in order to
animate a new one.

It would matter on how you rule "killing blow". Also, I don't have the skill card that grants Necromancy, so I'd have one question... mainly does it cost anything to activate? Requiring the hero to spend fatigue would be a dead giveaway.

If not, then *I* would probably rule that you would have to hit the monster with an attack to activate your Necromancy, but that's just my gut feeling on how the spirit of the rules is. (and we all know what happens when you go by that vs. the letter of the rules )

Either way, just make sure you're up-front about it with your players.

This depends on one of the answers to the new list of unanswered questions. The "killing blow" text is the same used in the encounter leader CT/100 coin reward text (but slightly different than that of the Lt death reward text). If the answer comes back "killing blow" = attack on that, then I assume the answer applies here as well.