Lunch in the Mural Room is a really nice addition to the itinerary! Our ABD group had a picnic beside String Lake. We didn't go to Jackson Lake Lodge at all.
Our family went back to the Tetons and Yellowstone when the kids were teenagers and stopped there to eat at the Pioneer Grill lunch counter. The views from the Mural Room and the porch out back were stunning!
That's great that you got to stay at the Old Faithful Inn and what an incredible view from your room! Our group stayed at the Snow Lodge, also not air-conditioned, although it was pretty cool the night we were there.