Oil - because particles of the non-halal food will break off while being fried, and a portion of those will most likely end up in otherwise halal food friend in the same oil.
The utensil/knife issue - if the utensil cuts through a piece of haraam material, some of that haraam material will loosely adhere to the utensil. When the unwashed utensil is used to cut otherwise halah food, that adhering haraam material will be deposited on the halaal food.