I would say no to adding wainscoting or shiplap, because wainscoting will cut your walls in half and make your ceiling height appear lower. And shiplap doesn't seem to go with the style of your cabinetry or countertops. You'd be better served by adding furnishings to the right back wall of the dining area. Maybe a major tall piece, television or bookshelves. Or if there is room, a seating area. Mid-level lighting would also be good in that back corner. Think function that works well for your family first and then find pieces that make life comfortable. A pair of paneled colorful drapes, hung from ceiling to floor, on either side of your Palladian Blue doors might look nice, but is not necessary. I like solar shades for patio doors, because it lets the light in and preserves the view when lowered, but can be fully opened to enter and exit.

I think you would want to keep paint colors light, since your countertop and appliances are dark. I don't see any reason why it would not make sense to match the underside of the soffit with your ceiling color, especially since your soffits extend far beyond your cabinetry in some areas. Think of it as a dropped ceiling rather than a soffit. I don't think you should to add additional trim to the soffits, because that will just make it fussy. I would not add backsplash tile since you are keeping the 4" backsplash and your cooktop is in the island. I think your 4" backsplash looks fine and is ideally functional behind the sink, since it is a perfect fit under the window stool.

Hard to see because of the shadows, but paint the soffits Simply White in the finish you used for cabinets and paint the ceiling Simply White, flat finish.

You don’t need a backsplash because, ostensibly, there is one.

I love Revere Pewter, I have Revere Pewter, and I have Palladian Blue.

However for your kitchen and the wall over your backsplash, I would go gray…make those cabinets shine.

No “ship lap”…too busy and most importantly it won’t be authentic ship lap.

On the dining wall, hang art on either side of doorway. I notice the ceiling lights reflect perfectly on the walls to showcase art.

Not suggesting this art, just using the photo for inspiration:

I am not usually a huge fan of white walls, but in your case I think I would paint the walls (and soffits) the same color as the cabinets, to allow the blue doors to really pop. No shiplap. Bring more of that blue color into the dining room with artwork or furniture pieces (I like Theresa's idea of something tall, like a hutch or bookshelf).

As for the kitchen, that 4" backsplash should pop off very easily (youtube it - a screwdriver and a few taps of a hammer is typically all it takes). I would remove it and put in a tile backsplash. Something light to brighten up the dark counters, maybe with some blue in it as well. If you have the budget, adding some additional lighting in the kitchen would also help.