Does the Traitor automatically get all items that a dead Hero had?
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.