I'm the Traitor and I've just killed a Hero who had two Items and an Omen card. Do I get to immediately pick up and use both the Items and the Omen?
If you kill the Hero on the Hero's turn (or during your monsters' turn), then you don't get to do anything with the items; you can only pick up items on your own turn. (Though you can steal an item on your opponent's turn in lieu of inflicting at least 2 points of damage, as normal. But then you didn't kill the Hero.)
If you killed the Hero on your turn, then the answer is both yes and no; per the official errata, an item can only be picked up or used on a given turn. You can pick up all of the items immediately, but you can't use any of them until a future turn. (This also prevents abuse from the Heroes; if there's an item you need to hold in order to make a Haunt-related roll, you can't pass around the item and make a Haunt-related roll with the item in the same turn.)
On a future turn, you can use all of the items. For items like the Armor that have a passive effect, they are effective immediately; it's only the items that you use during a turn (like the Bell or Medical Kit) that cannot be used during the turn you pick them up.