I am tired of backsplashes. All this beautiful kitchen needs is a rug.

Definitely agree with the island suggestion! For the backsplash, something rustic like a faded brick would juxtapose nicely with your modern kitchen, but all the way up to the ceiling. For additional storage, why not add 2 white box shelves across the entire length of your kitchen walls on top of the brick?

I wouldn't add a backsplash.
If the space is small keeping the same wall color gives you an open feeling instead of a chunky feeling.
You could swap the frame next to the stove for a really big colorful one.
I think that the colors should come from your accessories.

Is this your kitchen? I'd go for a humongous island.

Cabinets up to the ceiling! You will see big difference! Upper cabinets will make visually higher the ceiling.

The horizontal cabinets from Ikea work well for this problem.

I would get white cabinets for the uppers, so that they sort of blend into the ceiling. Get them as tall as will fit, keep it simple.

White cabinets and backsplash will keep you from subdividing the small space, creating as much continuity (as well as lightness!) as possible.

I think a large island would provide you with lots of extra storage and work surface. I would suggest a white subway tile with a grey grout and take it all the way up to the ceiling.