This is my paint color
Gray backsplash I would want

I know you're asking about back splashes, but I would be remiss if I didn't tell you to stop and rethink that granite. It is way too small of busy pattern and already looks very dated. I would hate to see you waste a lot of money on something that is already out of style. Have someone with some decorating sense help you before you spend money on a mistake like that.

I would love to but it’s just not in our budget. The only other option has a burgundy color in it.

I think your combination is fine. I'd do a warm white (something matched against the counter stone), but probably in ceramic rather than glass -- mostly because the latter can end up taking on a blue tinge when installed, which would probably give you the opposite effect that you intend.

I have often done gray glass with a counter like this, but that's in the context of white cabinets. with the darker wood, I think a light tile is the way to go (with matching grout), for more contrast, and a warmer white will help support your use of natural wood elsewhere.

Thank you you for your suggestion. Is this kind of what your talking about?

There are always other options in life. Find a discount granite place in your area and go and look through their granite sheets. I just redid a good sized kitchen in a stunning large movement granite piece for about $3,800 installed. I don't know what your budget is, but I don't understand the logic of knowingly buying something not good instead of finding better pricing and then saving for what you need. Do yourself a favor and rethink it.

From your photos, it looks like there is a warm or creamy element in your granite. I suggest a white subway tile that picks up that warm white. And not glass.

Should I do a matte or semi-gloss finish?

You can do any subway tile that is the right undertone to work with your granite. And it doesn’t have to be the standard 3x6. There are other sizes and also different layout patterns. Do not do a contrasting grout, as it will be too busy. But check out photos and article on Houzz - there are plenty that show different subway tile installations.