1) Yes, paint the brick white.

2) Nearly "any" tiled backsplash would work. Since the brick will be white, I suggest something like this:

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3) The backsplash should run the length of the counter and stop just beneath the upper cabinets.

I'd also paint it white to blend in w/the cabs. as for that vent hood, what kind of stove are are your doing? could you do a metal vent cover of some sort?

what does the rest of the kitchen look like? The tile choices above are good.

or you could do a solid back painted glass

Stove will be stainless steel Miele gas range.

Great this is a better perspective. Beautiful cabinetry, hardware and white cabinets above window. Painting the brick white with masonry paint that would seal it so it could be used as a back splash would be an option. You could also have a custom glass to protect and clean easy behind stove or stainless.

I would cover it with tile or stainless. It isn't the right color or texture for your style of kitchen.

Thanks so much for the great recommendations and insights!