Haiti has this 7.0 earthquake, I have not heard of damage at all in the Dominican Republic. Any idea why not?

There was a minor amount. But the media is focused on the main story.

And the reason why the damage was minor is the same one as why the rest of Haiti suffered very little: soil type. Much of Port-au-Prince is built on soft alluvium, which acts like Jell-o during a quake and magnifies the effects. But most of Hispaniola is mountainous, with lots of bedrock. Bedrock is the most stable type of soil to build on, and does not amplify the shaking.

There was some damage in the Dominican, but not much. Most of the damage was in parts of Haiti, including all the severe damage

Because the epicenter of the earthquake was so close to the surface (in geologic terms). And, the closer the epicenter is to the surface, the smaller the affected area and the more devestating the damage in that smaller area.

when one plate moves, it causes damage to it's side, when the other moves it causes damage to it's side.. more or less.

that's not always the case tho... depth, and upper layer composition of material play a part in damage too.

google 'quakes and check out some of the info .. for so powerful a force, it can be amazingly selective at times