I think a cleaner fresher look might be a nice look for the rooms.

How about some fab wallpaper for the staircase and some art in the DR

I would forget about the wainscotting. It will just make it look choppy. New paint and a bit of color in art work, etc, will make your room look so much nicer and more interesting. Right now, every picture, I have seen, of your house, is all browns and isnt very welcoming. I do like the flooring in your entry way, and the bit of walnut color on the stairway, but the walls look empty, and there is nothing in the way of any colors except those in the brown tones, which make it a bit boring.

IMO it will look odd to have wainscotting in various place. Having a full and consistent placement should look better. As for the tray ceiling it depends if you want to emphasize it or not. Tray ceilings are pretty dated at this point. So, if you don't want to emphasize it I would paint it the same color as the rest of the ceiling.

If you want wainscoting going up the stairs, but not in the foyer, yes it will look odd. Either do the entire foyer and stairs or don't do it at all. Properly done by a good trim carpenter will be fairly expensive, so it all depends on your budget. I think you'd get more for your money if you wallpapered the foyer/stairway/upstairs hallway.

Doing some sort of wainscoting in the dining room would be very nice. What do you mean by shaker style? The trim, doors and window casings look to be more Colonial in style, so you would need to install a wainscoting that goes with that style. Shaker is very plain but that's not what your trim is. Or change out all your trim and doors, which would be a waste of money.