You can check in (on Delta) at either end. However, if the hotel will drop you off at the International side, the experience is much better IMO.

You could take the train to the rental car area then bus to Int'l, but this would not be worth it.

I suspect the hotels will serve the Int'l side but I don't know for sure. Since either side is OK I would just ask when you get there if no one responds with better info.