I think everything looks great so far! I understand the fatigue though -- the decisions are seemingly endless. Can you show us pictures of where the barn door would be and wider shots of the areas? Given your fairly neutral color palette, you have so many options for a backsplash. What do you want? Neutral? An accent color? If you go for a white, make sure it matches the tone of your cabinets and I would choose an interesting shape -- herringbone, hexagons, etc.

I would not choose the tile you show because it has too much variation. You have a lot of texture already, which I love, so I would pick a simple, solid color tile, in a light gray or white (if you can find a white that doesn't conflict with your cabinets).

After seeing your additional pictures, I would paint the barn doors white.

What is the area with the stained/rustic wood on the wall? That could be another option for the barn doors (and would show less dirt/dust than white doors). White would look good though.