Do tomatoes love to grow downhill without cages if they do how do you build a raised bed on an existing slope that will not have an erosion problem?

Tomatoes generally do better with cages. Otherwise the tomatoes on the ground are more prone to bugs and rotting where they are on the ground. However, downhill shouldn't hurt them, but you'll want to make the raised beds with the slope, and try and cut them to be as flat as possible to help block erosion. Another way to help erosion is to put rocks on the higher part of the hill which will slow down the water.