The convention is, you paint the edges and faces etc, the same colour as the trim in the room it opens into, and, as mentioned, stain surfaces visible in the kitchen, when the door is closed. As for the mixture of wood/mdf framing, why bother? If you're going to stain one side, that overrides the other concern- just prime and paint the dining room side you don't want to stain. MDF is only used for painted door jambs etc in place of 'real' wood because it's cheaper, doesn't have knots/flaws etc.

Amy, I probably would paint the side door the same color as the siding. In our current home all side doors are the same color as the siding. Like your door, they're all out of sight from the front of the house. In our previous home, the side door was also out of sight from the front, but we painted it white which was our trim color.

I am doing the same as DLM. Mahogany door stained on outside and inside. My interior will have white trim. Wood floors and some stained wood stair parts, and other wood accents to pull it together. Or you can paint your interior door.

I love barn doors. But my style is more rustic glam/pottery barn. And I have cows. However, a barn door is a terrible choice as a bedroom door. It should go in a space where leaving the door open is fine. You don't really want your bedroom door open and you aren't going to slide the barn door all the time. For example, in my flip home (barn doors will still be in style when I sell next year!), I am putting a barn door between the main living space in the basement and the spare room/office. The door would normally be left open and the spot calls for it. In my new home, I am wanting to put one inside my master bedroom as a way to cover up my "coffee/wine bar" that is inset into the wall. Normally the barn door will be left open, but if I wanted to close it to hide the coffee bar, I could.