Showpiece kitchen you have there. Contact Alex Havens at Havens Metal for your custom copper hood. It will be gorgeous, high-quality, and one of the best values around if you live in the USA. I would not replace the backsplash until after the new hood is installed. Painting the sheetrock black would overpower the color scheme you have there. Copper is a much better solution to retain the light and color balance. I can understand your wife's wanting a more simple backsplash, but there are so many great tile companies out there for backsplashes, encourage her to really take her time with this and get it right with research and bringing samples into the space first.

If you can get wood to match the existing trim at the top of the hood, I think painting the sheetrock matte black (or almost black) is a good temporary solution. With the black ovens already on each side, I don't think the black hood will be too overwhelming. Probably will also depend on your new backsplash, so you might want to make that decision first.