No Shin-Tokyo station, just Tokyo Station. The "Shin-" just means "new" and is not short for "Shinkansen" Station.
Don't think we'll need to use JR trains or private trains to get around Kyoto. We're young and we love to walk everywhere. The Kyoto subway and a few taxis should take care of us. That said, if we buy a pass, we might as well put it to use
I too recommend taking the local bus network (there's a bus navi map you can pick up with most of the major sights and which bus lines will take you there), I often do a bus/subway combination if there's traffic, going to the nearest subway station then changing to a bus for the rest of the route.