I like wood interiors, but I prefer light colors. If they have to be mixed, I'd go with white oak and red oak battens. The red oak will kind of frame the boards. I've never heard of board and batten wainscoting. I thought that was reserved for barns. I'm kind of partial to random width shiplap myself, though that's a little informal for a dining room. For a dining room I'd rather see either bead board (still a little informal) or frame and panel. I guess the board and batten would go well with "Arts and Crafts" architecture though.
The house was built in 1914 and was not well maintained. We are working to make this home nicer than it would have been by just simply fixing plaster and painting (the kitchen was horrible!). It has some A&C touches to it so we are moving in that direction with the interior restoration. We have already done a gut reno of the kitchen/pantry/back hall, and of the attic. Now, the dining room, then the living room and other downstairs areas.
Shiplap is nice, but as you say, it is pretty informal looking, and more suited to a New England/costal/cottage style home. We've already put beadboard in the kitchen and bathroom, and, board & batten (B&B) is a defining feature in many dining rooms (and other rooms/hallways) of turn-of-the-last-century homes. The 1st floor has oak window/door casing, baseboards, crown molding, picture rail, doors, and some other trim, but I have not been able to positively identifying which oak it is as the old growth stuff looks a bit different from the newer growth oak, and in some respects it is not easy to zero in on white/red oak characteristics.
50 years ago, or so, the wood was painted white; we have stripped some of it off in the dining room, but not perfectly as there is paint in lots of grain/holes in the wood, still. We have been considering repainting the trim in the dining room white (Benjamin Moore's Mountain Peak White), and using poplar for the B&B and paint it along with the trim (except for the doors, crown molding, picture rail, and window casing/sash, which will get stained) for cost reasons. The wainscoting will not come in contact with the window trim, I have a design in mind to do that, but there is the option to go with all of the wood stained if we go with oak for the B&B, assuming the cost is not too crazy high.
