WHile we do not have any stained woodwork in our house (except entry doors), we do have our trim painted in different colors. I think it depends partially on the room transitions. What I mean is if your living room and dining room are joined without much of a transition, then it may look odd having a mix of stained and painted trim. However, if you cannot see the painted trim from any roon with stained trim, you should be OK.
As far as the doors are concerned, I have always used the following rule of thumb: close the door and look at it from each side. Whichever side you are looking at should be painted/stained to match that side of the room or hallway. That means painting/staining each half of the jamb and door differently. We even went as far as having different knobs on each side of the door.
Remember, you will never be able to see both sides of a door at once!