I'd suggesting keeping or adding some black somewhere beside the floor. Otherwise the contrast between your black floors (which I love, btw) and white everything else might be too stark.

I do love your current green as well.

What about bigger black hardware for the cabinets? I agree with keeping elements of black somewhere

No to painting the tile.

Will it be way too much white with white walls, white cabinets and white countertop?

Yes.

Will this tile backsplash paint well?

No. Please don't do that.

Paint the cabinets a soft green so it's not so white.

If it were me, I'd do a soapstone countertop and black handles/knobs.

I would remove it. Really, octagon and dot is a floor tile anyway. If you need to save money use a short backsplash of countertop material except between the cooktop and range hood.