The _cost_ to produce food goes up. Prices may or may not go up. Food producers are charging what people will pay, not what it costs them. Competition might be reduced when everyone’s input costs increase, but this principle still remains. If the market will bear higher food prices, and energy prices remain unchanged, then it’s simply the food producers that reap profits instead of energy companies.
Of course they do. Here's a simple example: if the price of energy goes up, the price of food and various goods goes up across the board.