What should I do for a backsplash and under counter overhang?

On the wall behind my cooktop I'm only planning on having this range hood. I may add open shelves in the future, but I prefer an open look. My question is what should I do for a backsplash? As you can see I like neutral colors. The countertop is white quartz btw. Maybe a carrara marble slab or carrara 3x6 tiles?

Under the counter is also an issue. Maybe the same material as the cabinets?

Any help is greatly appreciated...

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